Geometry of the Projectivization of Ideals and Applications to Problems of Birationality Remi Bignalet-Cazalet

Geometry of the projectivization of ideals and applications to problems of birationality Remi Bignalet-Cazalet

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Remi Bignalet-Cazalet. Geometry of the projectivization of ideals and applications to problems of birationality. Algebraic Geometry [math.AG]. Université Bourgogne Franche-Comté, 2018. English. NNT : 2018UBFCK038. tel-02004276

HAL Id: tel-02004276 https://tel.archives-ouvertes.fr/tel-02004276 Submitted on 1 Feb 2019

HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. These` de doctorat

de l’Universite´ de Bourgogne Franche-Comte´ préparéeàl’Institut de Mathématiquesde Bourgogne

Ecole´ doctorale Carnot Pasteur (ED553)

Doctorat de Math´ematiques

par R´emiBignalet-Cazalet

Geom´ etrie´ de la projectivisation des ideaux´ et applications aux problemes` de birationalite´

Directeurs de th`ese: Adrien Dubouloz et Daniele Faenzi

Th`esesoutenue le 24 octobre 2018 `aDijon

Composition du jury

Laurent Buse´ INRIA - Sophia Antipolis Examinateur Julie Deserti´ IMJ-PRG - UniversitéParis Diderot Examinatrice Adrien Dubouloz IMB - Universitéde Bourgogne Directeur de thèse Daniele Faenzi IMB - Universitéde Bourgogne Directeur de thèse Laurent Manivel IMT - UniversitéPaul Sabatier Rapporteur Lucy Moser-Jauslin IMB - Universitéde Bourgogne Présidente du jury Fransesco Russo DMI - Universitàdegli Studi di Catania Rapporteur Ronan Terpereau IMB - Universitéde Bourgogne Examinateur Institut de Mathématiquesde ´ Bourgogne Ecole Doctorale Carnot-Pasteur UMR 5584 CNRS Universitéde Bourgogne Universitéde Bourgogne Franche-Comté Franche-Comté UFR Sciences et Techniques 9 Avenue Alain Savary 9 Avenue Alain Savary BP 47870 21078 Dijon Cedex BP 47870 21078 Dijon Cedex 1

Al andar se hace camino y al volver la vista atr´as se ve la senda que nunca se ha de volver a pisar. Antonio Machado, Caminante no hay camino 2 3

En marchant se fait le chemin et c’est en se retournant que l’on peut contempler le sentier que l’on n’aura jamais plus l’occasion d’emprunter. Traduction personnelle 4 5

Remerciements

Au cours de ma thèse,j’ai pu bénéficierdu concours, du soutien ou de l’aide de personnes que je tiens àremercier. J’espèreque ces quelques lignes traduiront ma profonde gratitude envers eux. Mes premiers remerciements vont àmes deux directeurs de thèse, Adrien Dubouloz et Daniele Faenzi. Que de patience, que d’indulgence de leur part durant ces trois années! Je tiens àles remercier particulièrement pour leur disponibilité et leur exigence lors de toutes les phases rédactionnellesde mon doctorat. J’ai pu comprendre, grâceàeux, les passages obligésd’élaboration, de maturation et de diffusion d’une idéemathématique.J’ai beaucoup appréciéaussi les discussions passionnéesd’Adrien sur le fonctionnement du monde mathématiquequ’il m’a aidé àmieux envisager et la clairvoyance et les fulgurances de Daniele qui ont étédes sources constantes d’inspiration. Un grand merci aussi à • Laurent Manivel et Francesco Russo pour avoir rapportécette thèse.Leurs remarques ont permis de grandement améliorerle manuscrit initial et m’ont fait prendre conscience de questions que je n’avais pas encore clairement identifiées,notamment dans les chapitres 4 et 6, • Julie Désertiqui m’a aiguillésur cette thèseaprèsavoir encadrémon mémoire de master. Les thématiquesque j’ai abordéesdurant ces trois annéesont été trèsinfluencéespar ce travail initial et je la remercie d’autant plus d’avoir acceptéd’êtredans mon jury de thèse. • Laurent Busé,Lucy Moser-Jauslin et Ronan Terpereau pour avoir accepté de faire partie de mon jury de thèseet particulièrement Ronan dont j’ai énormément pu apprécierle dynamisme et l’engagement dans les activités mathématiqueset extra-mathématiquesdijonnaises. De manièregénérale,merci àla dynamique équipe de Géométriealgébrique de Dijon, dont FrédéricDégliseet Jan Nagel, pour m’avoir fait découvriret associé àtant d’activitésmathématiques,àJosé-Luispour sa bienveillance, àEmmanuel qui m’a mis le pied àl’étrierde l’enseignement àl’universitéainsi qu’àPeggy et Fran¸coispour leur aide précieusedans certaines de mes démarches administratives. Je tiens àremercier Charlie dont la fiabilitérésiste àtoute épreuve et ceux qui m’ont rendu la vie de doctorant àDijon plus agréable,les membres de l’IMB et toute son équipe administrative, les doctorants ou ex-doctorants de l’IMB, en particulier mes deux anciens co-brasseurs Antoine et Jérémy avec qui j’ai passéune première annéede thèseparticulièrement productive... Je tiens àciter Matilde, Vladimiro, Andrès,Anne, Mathias, Fabio dont j’ai énormément appréciéla compagnie entre autre dans des conférencesou écolesde maths. Un merci plus spécial àMatilde pour m’avoir accueilli dans cette belle ville de San Marin durant quelques jours si particuliers d’août.Plus généralement, je remercie tous mes amis, palois, palois désormaisparisiens et autres provinciaux parisiens, dijonnais ou anciens dijonnais ainsi que toute ma famille qui ont étéd’un soutien sans faille tout au long de mon doctorat.

Dijon, le 10 octobre 2018. 6

R´esum´egrand public

L’explosion récente des capacitésde calculs numériquesqui s’est produite au cours des trente dernières annéesa renverséles perspectives de recherches et développements. La simulation numériquepermet désormaisune estimation plus fiable et beaucoup plus rapide de résultatsautrefois inatteignables. Dans cette thèse,nous appliquons ce principe trèsgénéralàun domaine des mathématiques,la géométriealgébrique,qui concerne l’étudedes équations polynomiales. Cela nous permet de prévoir l’existence de configurations géométriquesinattendues apportant ainsi un point de vue original sur des su- jets classiques de géométrie.En plus de ces simulations, nous nous attachons àdémontrer mathématiquement ces phénomènesannoncésnumériquement. Nos résultatsconcernent aussi le développement de méthodes numériques visant àaméliorerdavantage les capacitéset la rapiditédes simulations dans ce domaine.

Large audience abstract

The recent growth in capacity of numerical calculus over the last thirty years has brought a major change in perspective regarding matters of research and development. Numerical simulation allows a more reliable and faster estimation of results unattainable beforehand. In this thesis, we apply this very general principle to a domain of mathematics, algebraic geometry which concerns the study of polynomial equations. This allows us to predict unexpected geometrical conﬁgurations bringing an original point of view to classical geometrical subjects. In addition of those simulations, we focus on proving mathematically those predicted phenomenons. Our results also concern the development of numerical methods aiming to further improve the capacity and the rapidity of simulations in this area. 7

R´esum´e

Dans cette thèse,nous interprétonsgéométriquement la torsion de l’algèbre symétriqued’un faisceau d’idéaux IZ d’un schéma Z définipar n+1 équations dans une variétén-dimensionnelle. Ceci revient àétudierla géométriede la projectivisation de IZ . Les applications de ce point de vue concernent en particulier le domaine des transformations birationnelles de l’espace projectif de dimension 3 au sujet duquel nous construisons des transformations birationnelles explicites qui ont le mêmedegréalgébrique que leur inverse, le domaine des courbes libres et presque-libres au sujet duquel nous généralisons une caractérisationdes courbes libres en étendant les notions de nombre de Milnor et de nombre de Tjurina. Nous abordons aussi le sujet des hypersurfaces homaloides, notre motivation initiale, au sujet duquel nous exhibons en particulier une courbe homaloide de degré5 en caractéristique3. La dernière application concerne le calcul de l’inverse d’une transformation birationnelle.

Mots clés: Géométriealgébrique,Algèbrecommutative, Théoriedes singularités, Transformations birationelles, Hypersurfaces homalo¨ıdes,courbes libres et presque libres, algèbrede Rees et algèbresymmétrique,Syzygies, Résolutions

Title of the thesis : Geometry of the projectivization of ideals and applications to problems of birationality

Abstract

In this thesis, we interpret geometrically the torsion of the symmetric algebra of the ideal sheaf IZ of a scheme Z deﬁned by n + 1 equations in an n-dimensional variety. This is equivalent to study the geometry of the projectivization of IZ . The applications of this point of view concern, in particular, the topic of birational maps of the projective space of dimension 3 for which we construct explicit birational maps that have the same algebraic degree as their inverse, free and nearly-free curves for which we generalise a characterization of free curves by extending the notion of Milnor and Tjurina numbers. We tackle also the topic of homaloidal hypersurfaces, our original motivation, for which we produce in particular a homaloidal curve of degree 5 in characteristic 3. The last application concerns the computation of the inverse of a birational map.

Keywords: Algebraic Geometry, Commutative algebra, Singularity theory, Bira- tional maps, Homaloidal hypersurfaces, free and nearly free curves, Symmetric and Rees algebra, Syzygies, Resolutions 8

Contents

Introduction 11 Contents of the manuscript ...... 17

1 Degrees of rational maps 23 1.1 Proj of a sheaf ...... 23 1.2 Algebraic cycles and Chow rings ...... 24 1.3 Rational maps ...... 25 1.3.1 The general setting...... 25 1.3.2 Construction with homogeneous polynomials ...... 27 1.3.3 Multidegree of a rational map and cycles in a Segre product . 27 1.4 The Cremona group ...... 31 1.4.1 First properties of the Cremona group...... 31

I Projectivization of an ideal sheaf 35

2 Torsion of the symmetric algebra 37 2.1 Projectivization and blow-up ...... 38 2.1.1 Projectivization of an ideal sheaf...... 38 2.1.2 Blow-up of an ideal sheaf ...... 40 2.2 Symmetric algebra versus Rees algebra...... 42 2.2.1 Local complete intersections...... 42 2.2.2 Torsion of the symmetric algebra...... 44

3 Resolution of the symmetric algebra 51 3.1 Algebraico-geometric background...... 52 3.1.1 Cohomological results about direct image sheaves ...... 52 3.1.2 The Eagon-Northcott complex ...... 55 3.1.3 Gorenstein rings and linkage ...... 58 3.2 Subregularity of the symmetric algebra...... 60 3.2.1 Generalities...... 60 3.2.2 Local resolution of the symmetric algebra...... 63 3.2.3 Graded free resolution of the symmetric algebra ...... 72 9

II Application to the study of rational maps 77

4 n-to-n-tic maps 79 4.1 Expected degrees and naive projective degrees...... 81 4.2 Quarto-quartics...... 84 4.3 Determinantal quinto-quintics...... 94 4.4 Higher degree and higher dimension ...... 97 4.4.1 Some families of P3 ...... 98 4.4.2 Incursion in higher dimension...... 100 4.5 Homaloidal complete intersections ...... 101

5 Free and nearly free sheaves of relations 103 5.1 Identiﬁcation of free and nearly free sheaves ...... 105 5.2 Classiﬁcation of curves by Tjurina numbers ...... 109

6 Zero-dimensional base locus 113 6.1 First and second naive degrees ...... 114 6.1.1 Importance of the subregularity of the symmetric algebra . . 116 6.1.2 Cohomological preliminaries...... 117 6.1.3 Proof of Theorem 6.1.1 ...... 118 6.2 Study of homaloidal hypersurfaces ...... 120 6.2.1 Measure of the diﬀerence between Rees and symmetric algebras120 6.2.2 Proofs of Proposition 19 and Proposition 20 ...... 124

7 Inverse of a birational map 129 7.1 State of the art to compute the inverse ...... 130 7.1.1 Inverse of the standard Cremona map ...... 132 7.1.2 Jacobian dual...... 133 7.2 Computation of the inverse ...... 134

Bibliography 141

Introduction

We introduce first the basic objects and topics we deal with and we present our contribution to these topics in the second section. Our objects of interest are rational maps between algebraic varieties over a field k. A rational map Φ : X 99K Y is an equivalence class of pairs hU, Φi where U is a nonempty open subset of X, ΦU is a morphism of U to Y , and where hU, ΦU i and hV, ΦV i are equivalent if ΦU and ΦV agree on U ∩ V . In general, Φ (more precisely a representative of Φ) is not defined everywhere on X and the closed subset Z of X where Φ is not defined is called the base locus of Φ. Letting U = X\Z so that the representative ΦU : U → Y of Φ is a morphism, the closure of the image of ΦU in Y does not depend on the representative of Φ and is called the image of Φ, denoted by Im(Φ). Given also a closed subvariety W −1 −1 of Y , the closure in X of the inverse image ΦU (W ), denoted by Φ (W ) is called the inverse transform of W under Φ. The graph ΓΦ of Φ is defined as the closure

{x, Φ(x), x ∈ U} ⊂ X × Y of the graph of the morphism ΦU : U → Y in X × Y (it does not depend on the representative of Φ as well). Since dominant rational maps Φ : X 99K Y and Ψ : Y 99K X can be composed, we are especially interested in the case when this composition has the identity map as representative. In this case, we say that Φ (and Ψ) is birational. Assuming that the varieties X and Y are irreducible of the same dimension and that Φ : X 99K Y is dominant (i.e. Im(Φ) is dense in Y ), we define the topological degree of Φ as the degree dt(Φ) = [k(X) : k(Y )] of the field extension of k(X) over k(Y ) the respective fraction fields of X and Y (cf Proposition 1.3.8 for a justification that the integer we just defined is indeed the actual topological degree of a morphism Φ : U → Y ). In this perspective, the property of Φ to be birational is equivalent to dt(Φ) = 1. Our main source of examples and applications comes from the situation where X = Y = Pn. In this case, a preferred representative of a rational map is defined by n + 1 homogeneous polynomials φ0, . . . , φn ∈ k[x0, ··· , xn] of the same degree δ without common component.

2 2 Example 1. Let τ : P 99K P be the map deﬁned by the polynomials x1x2, x0x2 and x0x1. The base locus Z of Φ is the union of the three points V(x1, x2) ∪

11 12 INTRODUCTION

V(x0, x2) ∪ V(x0, x1) (in the following, the notation V(s) for s a polynomial or more generally a section of a sheaf always stands for the zero locus of s in the ambient variety). 2 2 Moreover τ ◦ τ : P 99K P is deﬁned by the polynomials

(x0x1x2)x0, (x0x1x2)x1, (x0x1x2)x2.

But considering the map id deﬁned by the polynomials x0, x1, x2, we see that τ ◦ τ 2 and id coincide over P \ V(x0) ∪ V(x1) ∪ V(x2) . So, since rational maps are equivalence classes, a preferred representative of τ ◦ τ is the identity map deﬁned by x0, x1, x2. Actually, the map τ is quite special, at least because it is an involution (i.e. τ = τ −1). It is called the standard Cremona map.

More generally, X (respectively Y ) being a smooth subvariety of Pn, (respectively a smooth subvariety of Pm) we are interested in the multidegree of Φ. Roughly th letting k = dim(X) and i ∈ {0, . . . , k}, we can deﬁne a number di(Φ), called i projective degree of Φ, as follows

i −1 k−i di(Φ) = card H1 ∩ Φ (H2 )

i n k−i where H1 is a general (n − i)-plane of P and H2 is a general (m − k + 1)-plane of Pm. Less roughly, one has to use intersection theory, cf. Deﬁnition 1.3.12. The sequence dk(Φ), . . . , d0(Φ) is called the multidegree of Φ. When there is no ambiguity about the rational map Φ, we simply denote by (dk, . . . , d0) the multidegree of Φ.

n n Example 2. When Φ : P1 99K P2 is defined by n + 1 homogeneous polynomials φ0, . . . , φn ∈ k[x0, ··· , xn] of the same degree δ without common factor, the first projective degree d1(Φ) of Φ is the cardinality of the intersection of a hypersurface n P V( aiφi) of degree δ with n−1 general hyperplanes. Hence, by Bézout’stheorem, i=0 d1(Φ) = δ is the algebraic degree of the map Φ, i.e. the degree of each polynomials defining Φ.

3 3 Example 3. Let Φ : P1 99K P2 be a rational map of multidegree (1, d2, d1, 1) for a −1 given integer n. In particular Φ is birational of inverse Φ . But since ΓΦ = ΓΦ−1 , nd st we see that the 2 projective degree d2(Φ) of Φ, is also the 1 projective degree −1 −1 −1 d1(Φ ) of Φ , hence the algebraic degree of Φ . In greater generality, given an n n integer n ≥ 2 and a birational map Φ : P 99K P of multidegree (1, dn−1, . . . , d1, 1), −1 dn−1 is the algebraic degree of Φ .

Since the composition Φ1 ◦ Φ2 of two birational maps is birational, the set of birational maps has group structure whose group law is the composition of maps. This group is called the Cremona group of Pn, denoted by Bir(Pn) and its elements (birational maps) are also called Cremona maps. Let us present a problem related to the multidegree of the graph of a Cremona map Φ following [Dol11, 7.1.3]. Let n n Φ: P 99K P be a Cremona map and let (1, dn−1, . . . , d1, 1) be its multidegree. 13

Theorem 4. (L.Cremona, Cremona’s inequalities) For any n ≥ i, j ≥ 0,

1 ≤ di+j ≤ didj,

dn−i−j ≤ dn−idn−j.

See [Dol11, 7.1.7] for the proof of this result. Remark 5. [Dol11, 7.1.8] There are more conditions on the multidegree which follow from the irreducibility of the graph Γ of Φ. For example, if k is a ﬁeld of characteristic 0, by using the Hodge type inequalities we get the inequalities

2 di ≥ di−1di+1. These conditions on the multidegree lead to the following problem [SJ68]

Problem A. Let (1, dn−1, . . . , d1, 1) be a sequence of integers satisfying the Cre- mona inequalities and the Hodge type inequalities. Does there exist a Cremona map with this sequence as multidegree? As we will see, our work ﬁts in this question because we construct Cremona maps with given multidegree. The basis of these constructions is to study rational maps whose deﬁning polynomials are the maximal minors of a matrix. Let us explain in more details this situation.

Determinantal rational maps Let n ≥ 1 and M be a matrix of size (n + 1) × n whose entries are homogeneous polynomials in k[x0, ··· , xn] and such that all the entries of each column have the same algebraic degree. Then the n × n-minors of M define a rational map of Pn. These maps are called determinantal rational maps and, when birational, they are called determinantal Cremona maps. We are especially interested in determinantal Cremona maps under the influ- ence of the two articles [Pan99] and [DH17]. In [Pan99], the author describes in particular families of determinantal Cremona maps such as the families of determinantal cubo-cubic of P3. A Cremona map of this family is defined by the 3 × 3-minors of a 4 × 3 matrix with linear entries in k[x0, . . . , x3] (plus additional conditions) and it has multidegree (1, 3, 3, 1) from where the name determinantal cubo-cubic comes from. In [DH17], the two authors describes a family of determinantal quarto-quartic. A Cremona map of this family is defined by the 3×3-minors of a 4 × 3 matrix with two columns of linear entries and one columns of quadratic entries and it has multidegree (1, 4, 4, 1). We were particularly interested in con- structing other examples of determinantal maps, since, as we will see, such a study is well accessible by the objects we consider.

Homaloidal hypersurfaces and free divisors

We present now another type of Cremona map of Pn, those whose polynomials are the partial derivatives of a homogeneous polynomial f ∈ k[x0, ··· , xn]. Even if our results and research embrace eventually a more general point of view, namely, studying the presentation of the base ideal of any rational map, we emphasize that 14 INTRODUCTION tackling the domain of homaloidal hypersurfaces was our first motivation. Here the field k is an algebraically closed field of any characteristic and we let n ≥ 1 be an integer.

Deﬁnition 6. Let f ∈ k[x0, . . . , xn] be a homogeneous polynomial and for all ∂f th i ∈ {0, . . . , n} we let fi = be the partial derivative of f with respect to the i ∂xi variables. We call the rational map

n n Φf : P P x f0(x): ... : fn(x) the polar map of the hypersurface F = V(f) deﬁned by f in Pn. The topological th degree of Φf , i.e. its n projective degree, is called the polar degree of F and the ideal of the base locus of Φf is called the jacobian ideal of F . The hypersurface F or the polynomial f is called homaloidal if Φf is birational.

We emphasize that if f has degree d, the partial derivatives fi have degree d−1 (or are 0). However, the algebraic degree of Φf may not be equal to d − 1 since we have to remove common factors of the fi in order that a representative of Φf has a codimension at least 2 base locus. This is the case in particular when f is not square α1 αm free, that is when one of the exponents αi in the decomposition f = q1 . . . qm into irreducible homogeneous polynomials qi ∈ k[x0, . . . , xn] is stricly greater than 1. Stating that f is square free is equivalent to set that the hypersurface F = V(f) is reduced. As examples of homaloidal hypersurfaces, we have the foundational result of I.V.Dolgachev classifying the reduced homaloidal curves assuming the base ﬁeld k is the ﬁeld of complex numbers C. Theorem 7. [Dol00, Theorem 4] The only complex reduced homaloidal curves are the smooth conics, the unions of three general lines and the unions of a smooth conic with one of its tangent.

Without giving the complete proof of this result, let us explain it in the following paragraphs. Given a polar map Φf = (f0 : ... : fn), the base locus Z = V(f0, . . . , fn) of Φ is precisely the singular locus of F , that is, the locus where the tangent space of F has bigger dimension than expected. When we consider curves in P2, this singular locus is necessarily 0-dimensional. Generalising to a hypersurface having only 0-dimensional singularities, one number permits to classify those singularities, the Milnor number:

Deﬁnition 8. [Mil68] Let Z be the 0-dimensional singular locus of a hypersurface F = V(f) of Pn and let z ∈ Z. Via a change of coordinates, suppose that z = (1 : 0 : ... : 0). Set g[ ∈ k[x1, . . . , xn], the usual deshomogeneisation of a homogeneous polynomial g ∈ k[x0, ··· , xn] in the chart {x0 6= 0}. The local Milnor number at z, denoted by µf (Z, z), is deﬁned as

∂f[ µf (Z, z) = length Okn,z/((f[)1,..., (f[)n) where (f[)i = . ∂xi 15

P The global Milnor number of F , denoted by µf (Z) is the sum µf (Z, z) over all z ∈ Z.

2 n Example 9. Let f = x0(x1 + x0x2) ∈ k[x0, x1, x2] and F = V(f) ⊂ P be the union of a smooth conic with one of its tangent. The singular locus Z is 2 2 equal to V(x1 + 2x0x2, 2x0x1, x0) so F is singular at the point z = (0 : 0 : 1). Hence, on the chart {x2 6= 0}, we have to compute the length of the module 2 k[x0, x1]/(x1 +2x0, 2x0x1). As a module, this quotient is only generated by 1, x0, x1 2 1 2 2 since, for example x0 ≡ − 2 x1x0 ≡ 0 mod (x1+2x0, 2x0x1). Hence µf = µf (Z, z) = 3.

When k = C, assuming that this singular locus is ﬁnite, the following relation is established by A.Dimca and S.Papadima [DP03]:

Theorem 10. Let f ∈ C[x0, . . . , xn] be a square free homogeneous polynomial of n degree d and let Φf be the polar map of F = V(f) ⊂ P . Assuming that F has ﬁnite base locus, we have:

n dn(Φf ) = (d − 1) − µf (Z). (0.0.1)

Let us explain why the assumption that f is square free is important here. Since the singular locus of F = V(f) is 0-dimensional and f is square free, the polynomials fi defining Φf cannot have a common factor (or else the singular locus of F would have codimension 1), hence the algebraic degree of Φf is d − 1. So let us explain the relation (0.0.1) as follows. As we will see in more details in Example 1.3.9, computing the topological degree of Φf is the same as computing the degree of the intersection of n pull back V(ai0f0+...+ainfn) of general hyperplanes n V(ai0y0 + ... + ainyn) in P after removing the points in the base locus from this n intersection. By Bézout’stheorem, the intersection ∩ (ai0f0 + ... + ainfn) has i=1V n degree (d − 1) since the polynomials defining Φf have degree d − 1. To compute n the topological degree of Φf , it remains to subtract to (d − 1) the multiplicity of the points in Z. Among all the numbers that can be attached to the base locus of Z, Theorem 10 states that we have actually to subtract the number µf (Z) to (d − 1)n.

Example 11. As an example, let us focus on the list of complex homaloidal curves in Theorem7.

• a smooth conic F = (f) ⊂ 2 veriﬁes µ = 0 so d (Φ ) = (2 − 1)2 = 1. V PC f t f • up to a projective change of coordinate, the union of three general lines is the zero locus of f = x0x1x2. It has 3 singular points each one verifying 2 µf (Z, z) = 1. Hence dt(Φf ) = (3 − 1) − 3 = 1. • as we saw in Example9, the global Milnor number of the union of a smooth conic with one of its tangent is equal to 3. Hence dt(Φf ) = 1. From the foundational classiﬁcation of I.V.Dolgachev of reduced complex homaloidal plane curves, the questions bifurcate in several directions for which we present a non-exhaustive list of results. 16 INTRODUCTION

Homaloidal hypersurfaces in higher dimension

Over the field C, we saw that there are three types of homaloidal curves in P2. Concerning the situation in higher dimension, let us first mention that general constructions of homaloidal hypersurfaces existed. For instance, we refer to [Man13, Theorem 21] for general results about the existence of homaloidal hypersurfaces in higher dimension. In addition to the construction of examples, one specific problem was to establish if given n ≥ 3, there exist homaloidal hypersurfaces of Pn of any degree. In [CRS08], the three authors answer positively to this question by producing explicit families of homaloidal hypersurfaces of arbitrary degree. More precisely:

Theorem 12. [CRS08, Theorem 13] For every n ≥ 3 and for every d ≥ 2n − 3 there exists homaloidal hypersurfaces F of Pn of degree d.

Homaloidal hypersurfaces with isolated singularities In [DP03, 3], A.Dimca and S.Papadima conjectured that given n ≥ 3, any complex homaloidal hypersurfaces of n must have a singular locus of codimension at most PC n − 1, i.e. in particular, cannot have a singular locus consisting of isolated points. This conjecture was then proven by J.Huh in [Huh14]. In a recent paper, D.Siersma, J.Steenbrink and M.Tib˘arproved that, more generally, there are restrictions to the existence of hypersurfaces with 0-dimensional singular locus and small topological degree. More precisely

Theorem 13. [SST18, Theorem 1.4] For any integer k ≥ 2, let Kk denote the set of pairs of integers (n, d) with n ≥ 2 and d ≥ 3, such that there exists a projective hypersurface V in Pn of degree d with isolated singularities and polar degree k. Then Kk is ﬁnite for any k ≥ 2.

Homaloidal hypersurfaces in positive characteristic Another domain concerning homaloidal hypersurfaces is to study the problem over other fields than C and in particular over algebraically closed fields of positive characteristic. As one can compute, except from a field of characteristic 2, the three plane curves of Theorem7 are still homaloidal. So let us state the problem as follows:

Problem B. Over an algebraically closed ﬁeld k of positive characteristic, are there other homaloidal curves than the ones in Dolgachev’s classiﬁcation?

Another problem in this direction is the following. It was noticed by A.V.D´oria, S.H.Hassanzadeh and A.Simis [DHS12] that a common property of the three complex homaloidal curves is that their singular locus Z is a local complete intersection at each of its points. This means that each localisation OZ,z of the structure sheaf of Z is generated by two elements even if it is generated globally by the three partial derivatives (as we will see, this property is equivalent to the fact that Milnor and Tjurina numbers of the curves coincide). CONTENTS OF THE MANUSCRIPT 17

Problem C. [DHS12, Question 2.7] Let f ∈ k[x0, x1, x2] be a square free homogeneous polynomial whose polar map is birational. Is the singular locus Z of F = V(f) locally a complete intersection at its points?

The reduction problem in positive characteristic In the spirit of studying the difference between characteristic zero and positive α1 αm characteristic, we also consider the following reduction problem. If f = q1 . . . qm is not square free, or equivalently if F is not reduced, the polar map Φf is defined by the mobile part of the linear system generated by f0, . . . , fn, see the explanation just after Definition6 about this problem. Over the field of complex numbers, it was established by A.Dimca and S.Papadima [DP03] that Φf is birational if and only if so is the polar map Φfred associated to fred = q1 . . . qm. Over a field of positive 2 characteristic, this equivalence trivially fails: in characteristic 2 for f = x yz,Φfred is birational whereas Φf is not even dominant. This leads to the following problem.

Problem D. Over a ﬁeld of positive characteristic, given Φf dominant, is it birational if and only if so is Φfred ?

Contents of the manuscript

n n Given a rational map Φ : P1 99K P2 deﬁned by n + 1 homogeneous polynomials φ0, . . . , φn ∈ k[x0, . . . , xn] of degree δ without common factor, we consider a locally free presentation of the base ideal IZ of Φ:

m M n+1 ⊕ O n (−ai) O n IZ (δ) 0 (0.0.2) i=1 P1 P1

n n with ai ≥ 1 for all i ∈ {1, . . . , m}. It deﬁnes an ideal sheaf I on P1 × P2 X generated by the entries of the row matrix y0 . . . yn M where y0, . . . , yn are n n the coordinates of P2 (x0, . . . , xn being the coordinates of the ﬁrst factor P1 ). Actually, the scheme X = V(IX) is the projectivization P(IZ ) of the ideal IZ n n embedded in P1 × P2 and contains the graph Γ of Φ as an irreducible component. We have thus the following commutative diagram

n n P1 × P2

p1 X p2

π1 π2 Γ σ1 σ2

n n P1 P2 Φ where p1 (resp. p2) is the ﬁrst projection (resp. second projection) and respectively π1 and σ1 (resp. π2 and σ2) are the restriction of p1 (resp. p2) to respectively X and Γ (resp. X and Γ). 18 INTRODUCTION

Our problem in this context is to relate the multidegree of Φ with the geometry of X. As we will explain, this is studying the difference between the symmetric and the Rees algebras of IX and, in this perspective, this is a classical problem of commutative algebra, see [Vas05] for an introduction to this point of view or [BCJ09] and [BCS10] for results in this direction. Chapter1 is dedicated to explain the background and the framework of the manuscript in a more detailed way and to define the multidegree of rational maps. In Chapter2 , we focus more precisely on the definition of the projectivization P(IZ ) of an ideal sheaf. We present also geometric properties of X. In Chapter3 , we are especially interested in the resolution of the ideal IX. Let us explain why. Define E as the image sheaf of the presentation matrix M in (0.0.2). We call E the sheaf of relations of IZ . If E is locally free of rank n then X is 0 n n ∗ ∨ ∗ the zero locus of a global section s ∈ H × , p (E )⊗p O n (1) . This gives to P1 P2 1 2 P2 the push forward p1∗ interesting cohomological properties with respect to a locally free resolution of IX. For instance, if E is split as a direct sum of line bundles, n n then X is a complete intersection in P1 × P2 . Hence the ideal sheaf IX is resolved by the Koszul complex associated to the generators of IX (see Definition 2.2.5 for the definition of the Koszul complex) and, as we will see, this implies that the multidegree of X, which we call the naive multidegree of Φ, is computed by the degree of the Chern classes of E. The result in Chapter3 is that even if E is not locally free, but assuming instead that Z = V(IZ ) is zero-dimensional, IX has a locally free resolution closed enough to a Koszul complex, namely:

Proposition 14. Assuming that Z = V(IZ ) is zero-dimensional and denoting n n P = P1 × P2 and ξ for the ﬁrst Chern class of OP(0, 1), a locally free resolution of IX reads

0 Gn+1 Gn ... G2 G1 IX 0

i ∗ where Gi = ⊕ p1 ∗ Tij ⊗ O (−jξ) when i ∈ {1, . . . , n} and Gn+1 = p1Tn ⊗ O (−ξ) j=1 P P n for some locally free sheaves Tij and Tn over P1 . Chapter6 is dedicated to the application of this result, namely, when Z is zero-dimensional, the naive multidegree of Φ is still computed by the length of a cosection of E (which are the analogues of Chern classes when E is not locally free). More generally, PartII is dedicated to the applications of considering the projectivization of the base ideal sheaf of a rational map. We emphasize that the dichotomy between the case where E is locally free and the case it is not locally free is structuring in our work and PartII reﬂects this dichotomy. In Chapter4 and Chapter5 , we focus on the case where E is locally free. More precisely, in Chapter4 , we consider the case where E is split. As we will explain, this is equivalent to the fact that IZ is the ideal of maximal minors of M, so Φ is a determinantal rational map. As we mentioned in Theorem4, the multidegree of a Cremona map veriﬁes the Cremona inequalities but it is not known if given a sequence (d0, . . . , dn) verifying Cremona inequalities, there exists a Cre- n n mona map P 99K P . In Chapter4, we construct examples of rational map with given multidegree by analysing the scheme X (or, equivalently by analysing the matrix M). It is in this way that we tackle ProblemA. Moreover, we construct CONTENTS OF THE MANUSCRIPT 19

3 3 −1 birational maps Φ : P1 99K P2 whose inverses Φ have the same algebraic degree than Φ. For simplicity we call these maps n-to-n-tics by analogy with the more common denomination cubo-cubic and quarto-quartic in degree 3 and 4. Recall from Example3 that the second multidegree d2 of Φ is the algebraic degree of Φ−1. Hence n-to-n-tic maps are the maps with multidegree (1, n, n, 1). This work is motivated by the two articles [Pan99] and [DH17]. One of our result in this topic is as follows:

Proposition 15. Over P3, let Φ be the determinantal map deﬁned by the 3 × 3 minors of the following matrix:

 2 2 x0 x2 + x3 x0x2 + x0x1x2 + x0x3 2 3x0 + x1 x2 + 2x3 x1x3 + x1x2x3  M =  2   x0 + x1 x2 x1x2 + x0x1x3  2 2 x0 + 2x1 x3 x0x3 + x1x3 then Φ is a quinto-quintic (i.e. has multidegree (1, 5, 5, 1)). Chapter5 and Chapter6 deal with the case where the base locus Z is zero dimensional. In the polar case, in addition to Milnor numbers, we can attach also the Tjurina number to a 0-dimensional singularity of a projective hypersurface. Deﬁnition 16. Let Z be the 0-dimensional singular locus of a hypersurface F = V(f) of Pn and let z ∈ Z. Via a change of coordinates, suppose that z = (1 : 0 : ... : 0). The local Tjurina number at z, denoted by τf (Z, z) is deﬁned as

∂f[ τf (Z, z) = length Okn,z/(f[, (f[)1,..., (f[)n) where (f[)i = . ∂xi P The global Tjurina number of F , denoted by τf (Z) of F , is the sum τf (Z, z) over all z ∈ Z. Milnor and Tjurina numbers have very close deﬁnitions and given the 0-dimensional singular locus Z of a hypersurface F = V(f), they verify the inequality

τf (Z) ≤ µf (Z) but, as we will see, this inequality is strict in general. Anticipating on Chapter6, n we can sum up our work by a study of the number dn(Φf ) = (d − 1) − τf which we call the nth naive projective degree or naive topological degree and a study of the difference µf (Z) − τf (Z). 2 2 In Chapter5 , we focus on the case of rational maps P 99K P . It is a result following from [Har80] that E is locally free of rank 2. Let us explain also our initial motivation for studying this problem. Let f ∈ k[x0, x1, x2] be a square free polynomial of degree d and let I be the jacobian ideal sheaf of f, i.e. the ideal ∂f sheaf generated by the partial derivatives fi = of f. Letting E be the sheaf of ∂xi relations of I, the curve F = V(f) is free if E is split (following the definition in 2 2 [Dim15]). If E is split and E'OP (−d1) ⊕ OP (−d2) then F is said to be free of exponent (d1, d2). A result of A.A. du Plessis and C.T.C.Wall in [dPW99] identifies in particular complex curves F of a given degree d with maximal possible global 20 INTRODUCTION

Tjurina number (see Deﬁnition 16 for the deﬁnition of Tjurina numbers). These are the free curves of exponents (1, d − 2). But, as we will explain, the data of the global Tjurina number of F is equivalent to the data of the second Chern class of E via the relation

2 c2(E) = (d − 1) − τf (Z). Hence the identification of curves with the highest global Tjurina number is equivalent to identifying the curves such that c2(E) is the smallest possible. In Definition 2.2.21, we define a generalised Tjurina number for any ideal sheaf 2 2 I of P generated by three global sections of OP (δ). This is just the length of the scheme V(I). So for us, a main motivation is to elaborate a similar criterion to split the sheaf of relations E. One result in this chapter is as follows:

Theorem 17. Let I be an ideal sheaf over P2 generated by three global sections 2 φ0, φ1, φ2 of OP (δ) and let E be the sheaf of relations of I (recall that E is defined 3 as the kernel of the evaluation map O 2 → I(δ)). Then −c1(E) ≤ c2(E) + 1 and P equality holds if and only if E is free of exponents 1, c2(E) . We emphasize that Theorem 17 is precisely a generalisation of the former result in [dPW99] since we identify free sheaves of exponents 1, c2(E) with sheaves of relations with the smallest second Chern class possible. The second part of this chapter is the classification of the reduced complex plane curves with respect to the second Chern class of their sheaf of relations, that is, we classify curves of degree 2 d and c2(E) = (d − 1) − τf . In Chapter6 , we consider the case where Z is zero-dimensional. In this case E is not locally free in general. Recall that the nth naive projective degree of Φ is n P the length of the intersection of n general generators λiφi of IZ where λi ∈ k i=0 for any i ∈ {0, . . . , n} after removing the points already in Z (see Section 1.3). Hence the nth projective degree of Φ is the length of the scheme V(s) defined in the following exact sequence:

n O n I (δ) O P1 Z V(s) 0

n where the morphism O n → IZ (δ) is general. Now consider this morphism in the P1 following commutative diagram:

n n O n = O n P1 P1

m M n+1 0 E = ⊕ O n (−ai) O n IZ (δ) 0 i=1 P1 P1

O n O P1 V(s) 0

0 0 CONTENTS OF THE MANUSCRIPT 21

n So V(s) is also the support of the cokernel of the map E = ⊕ O n (−ai) → O n . i=1 P1 P1 In other words, the class [V(s)] of V(s) in the Chow ring of Pn is the support of a cosection of E. The second way to compute this projective degree is to consider the projec- n n tivization X of IZ in P1 × P2 and to compute its decomposition in the Chow ring n n of P1 × P2 (see Section 1.2 for these deﬁnitions). The problem of this chapter is to establish if those two ways of computing the naive projective degrees, namely considering cosections of E or the projectivization X, always coincide even if E is not locally free. We summarize our result as follow, see Theorem 6.1.1 for a more precise statement.

Theorem 18. In the case when the base locus Z is 0-dimensional, the length of n+1 the zero scheme of a cosection of the kernel E of the evaluation map O n → IZ (δ) P1 is equal to the nth naive projective degree.

We answer also positively ProblemB and negatively ProblemC and ProblemD.

2 2 2 Proposition 19. The curve F = V (x1 + x0x2)x0(x1 + x0x2 + x0) is homaloidal if and only if the base ﬁeld k has characteristic 3, in which case the inverse of the polar map is

2 2 3 4 3 2 3 4 2 3 Ψ = (x1x2 + x0x2 + x2 : −x1x2 − x0x1x2 − x1x2 : −x1 − x0x1x2 + x0x2) Proposition 20. Let k be an algebraically closed ﬁeld of characteristic 101.

3 2 50 3 2 51 (i) The curve V z(y + x z) has polar degree 2 whereas V z (y + x z) has polar degree 1.

3 2 2 3 (ii) The curve V (y +x z)(y +xz) has polar degree 5 whereas the curve V (y + x2z)31(y2 + xz)4 has polar degree 3.

In Chapter7 , the goal is to provide a numerical way of computing the inverse −1 n n Φ of a rational map Φ : P 99K P using the geometry of the projectivization X of the base ideal sheaf IZ of Φ. We especially focus on ﬁnding the inverse of the 2 2 2 polar map of f = (x1 + x0x2)x0(x1 + x0x2 + x0) in characteristic 3.

Chapter 1

Degrees of rational maps

As we saw in the introduction, different degrees can be attached to a rational map and in particular the projective degrees. In this chapter, before stating our framework about rational maps, we will define these degrees. We end this chapter by a refined presentation of the Cremona group and by answering negatively the following problem (see Subsection 1.4.1 for its motivation).

Problem E. Is any Cremona map of P2, of algebraic degree stricly greater than 1, determinantal?

1.1 Proj of a sheaf

We mostly follow [Har77, II.7] for this background. We only come back to the relative notion of Proj of a sheaf of graded OX -algebras and we refer to [Har77, II.2] for the deﬁnition of the Proj of a ring R. The general setting is as follows:

X is a smooth quasi-projective variety and S is a positively graded sheaf of graded OX -module which has a structure of a sheaf of graded (4) OX -algebras. Thus S ' ⊕d≥0Sd where Sd is the homogeneous part of degree d. We assume that S0 = OX , that S1 is a coherent OX -module, and that S is locally generated by S1 as an OX -algebra.

Construction. For each open subset U = Spec(A) of X, let SU be the graded A-algebra Γ(U, SU ). Then, the schemes Proj(SU ) together with their morphisms Proj S(U) → U glue to give a scheme Proj S together with a morphism π : Proj S → X such that for each open aﬃne U ⊂ X, π−1(U) ' Proj S(U). As such, Proj S has an invertible sheaf O(1).

Example 1.1.1. Over X, let S be the polynomial algebra S = OX [y0, . . . , yn]. For n each open subset U = Spec(A) of X, SU is equal to A[y0, . . . , yn] so Proj SU = PA. n The morphisms PA → Spec(A) are those associated to the canonical morphisms A → A[y0, . . . , yn]. n n The scheme Proj S is denoted by PX or X ×k Pk . Lemma 1.1.2. [Har77, II.7.9] Let S be a sheaf of graded algebras on X as in (4). Let L be an invertible sheaf on X, and deﬁne a new sheaf of graded algebras

23 24 CHAPTER 1. DEGREES OF RATIONAL MAPS

0 0 d 0 S = S ⊗ L by Sd = Sd ⊗ L for each d ≥ 0. Then S satisﬁes (4) and there is a natural isomorphism φ : P 0 = Proj S0 → P = Proj S, commuting with the projections π and π0 to X, and such that

∗ 0∗ OP 0 (1) = φ OP (1) ⊗ π L. Definition 1.1.3. Let X be as in( 4) and let G be a coherent sheaf on X. We define the associated projective space bundle P(G) as follows. Let S = Sym(G) be the symmetric algebra of G. It verifies S = ⊕d>0 Symd(G). Then S is a sheaf of OX -algebras satisfying( 4) and we define

P(G) = Proj S.

As such, it comes with a natural projection π : P(G) → X. n+1 n Example 1.1.4. If G = OX , then S = OX [y0, . . . , yn] and P(G) is PX as in Example 1.1.1. More generally, let G be a locally free coherent sheaf of rank n + 1. −1 n Given an open set U of X trivialising G, we have that π (U) ' PU so P(G) is a ”relative projective space” over X. Proposition 1.1.5. [Har77, II.7.11] Let X be as in (4), G be a locally free coherent sheaf over X and π : P(G) → X be as in Deﬁnition 1.1.3. Then:

(a) if rank G ≥ 2, there is a canonical isomorphism of graded OX -algebras S' ⊕l∈ π∗ O(l) , with the grading on the right hand side given by l. In partic- Z ular, π∗ O(1) = G, (b) there is a natural surjective morphism π∗G → O(1).

1.2 Algebraic cycles and Chow rings

Let k be an algebraically closed field and let X be any variety over k i.e. here integral separated scheme over a field k. Definition 1.2.1. A cycle of codimension r on X is an element of the free abelian group Zr(X) generated by the closed irreducible subvarieties of X of codimension P r. So we write a cycle as Y = niYi where the Yi are closed subvarieties of X of codimension r. Sometimes it is usefull to speak of the cycle associated to a closed subscheme. If Z is a closed subscheme of pure codimension r, let Y1,...,Yt be those irreducible components of Z which have codimension r, and define the cycle P associated to Z to be niYi where ni is the length of the local ring OZ,Yi of the general point ui of Yi on Z. We suppose now that X is a smooth quasi-projective variety of dimension n. We refer to [Har77, II.6 and A.1] for the precise definition of rational equivalence n on Z(X) = ⊕ Zr(X). r=0 Definition 1.2.2. For each r, we let CHr(X) be the group of cycles of codimension r on X modulo rational equivalence. We denote by CH(X) the graded group n r ⊕r=0 CH (X) and given Z a subvariety of X, we denote by [Z] its class in CH(X). 0 r Note that CH (X) = Z and that CH (X) = 0 if r > n. 1.3. RATIONAL MAPS 25

We refer to [Har77, Axiom 1 to Axiom 11] and [Har77, Theorem 1.1] for the properties and results making CH(X) into a graded ring, called the Chow ring of X. The product is given by the intersection:

CHi(X) × CHj(X) CHi+j(X)

([Z1], [Z2]) [Z1 ∩ Z2]

assuming that Z1 ∩ Z2 has the expected dimension.

Proposition 1.2.3. [Har77, 2.0.1] Given n ∈ N\{0}, CH(Pn) ' Z[H]/(Hn+1) where H is the class of any hyperplane. In other words, any subvariety of degree d and codimension r in Pn is rationally equivalent to dHr. Before applying the previous notions for multidegree, let us deﬁne the Chern classes following [Har77, 3] One properties of the intersection product on the Chow ring is as follows:

Property 1.2.4. [Har77, Axiom 11] Let E be a locally free sheaf of rank r on X, 1 let P(E) be the associated projective bundle and let ξ ∈ CH P(E) be the class of the divisor corresponding to OP(E)(1). Let π : P(E) → X be the projection. Then ∗ 2 r−1 π makes CH P(E) into a free CH(X)-module generated by 1, ξ, ξ , . . . , ξ . Deﬁnition 1.2.5. Let E be a locally free sheaf of rank r on a nonsingular quasi- th projective variety X. For each i = 0, 1, . . . , r, we deﬁne the i Chern class ci(E) ∈ i CH (X) by the requirement c0(E) = 1 and

r X i ∗ r−i (−1) π ci(E)ξ = 0 i=0

r in CH P(E) . We refer to [Har77, C1 to C7] for the properties of the Chern classes but we insist on one interpretation that we will use.

Proposition 1.2.6. [Har77, A.C6] Let E be a locally free sheaf of rank r on a nonsingular quasi-projective variety X. If the dependency locus W , i.e. the common zero locus, of t global section has codimension r − t + 1 the class [W ] of W is equal r−t+1 to cr−t+1(E) in CH (E).

1.3 Rational maps

1.3.1 The general setting Let us ﬁrst give the following deﬁnition extracted from [Har77].

Deﬁnition 1.3.1. Let X,Y be varieties. A rational map Φ: X → Y is an equivalence class of pairs hU, ΦU i where U is a nonempty open subset of X,ΦU is a morphism of U to Y , and where hU, ΦU i and hV, ΦV i are equivalent if ΦU and ΦV agree on U ∩ V . 26 CHAPTER 1. DEGREES OF RATIONAL MAPS

When the target variety Y is the projective space Pr we have the following setup (as a convention, we emphasize that given a vector space V , we denote by P(V ) the space of hyperplanes of V ). Let n = dim(X), L be a line bundle over X and V be a non-zero vector subspace of H0(X, L). We can define a rational map ΦU on X by sending a point x ∈ X to the hyperplane Hx = {s ∈ V, s(x) = 0} where s(x) is the evaluation of the section s at the point x. Of course, this map is only defined over the points x such that there exists s ∈ V verifying s(x) 6= 0. The open subset U over which the map ΦU is defined is precisely the set of these points. Given the basis (φ0, . . . , φr) of V, we see that Φ sends a point x ∈ X at r which it is defined to the point φ0(x): ... : φr(x) ∈ P(V) ' P . Conversely, r given a rational map Φ : X 99K P , we can find a representative hU, ΦU i of Φ such that codim(X\U) > 1. Then hU, ΦU i defines a line bundle LU over U extending in a unique way on X (see also [Dol11, 7] for this construction). 0 Consider the natural evaluation map of sections of L, ev :V ⊗OX → L. It is equivalent to ∨ ev : V ⊗L → OX.

The image of ev is a sheaf of ideals IZ in OX , called the base ideal. Its support Z = V(IZ ) is a closed subscheme in X called the base locus of Φ. So letting Di = V(φi) the scheme of zeros of the section φi for i ∈ {0, . . . , r}, we have that Z is the scheme theoretic intersection D0 ∩ ... ∩ Dr in X.

r Deﬁnition 1.3.2. Let Φ : X 99K P be a rational map with base locus Z and r denote U = X\Z so that the representative ΦU : U → P of Φ is a morphism. r The closure of ΦU in P does not depend on the representative of Φ and is called the image of Φ, denoted by Im(Φ). Given also a closed subvariety W of Pr, the −1 −1 closure in X of the inverse image ΦU (W ), denoted by Φ (W ) is called the inverse transform of W under Φ.

r Deﬁnition 1.3.3. Let Φ : X 99K P be a rational map and let Y be the image variety of Φ. Denoting Z the base locus of Φ and U = X\Z, the graph Γ of Φ is deﬁned as the closure {x, Φ(x), x ∈ U} ⊂ X × Y of the graph of the morphism ΦU : U → Y in X × Y (it does not depend on the representative of Φ).

Since dominant rational maps Φ : X 99K Y and Ψ : Y 99K X can be composed, we are especially interested in the case when this composition has the identity map as representative.

Definition 1.3.4. Let X be a quasi-projective variety over a field k and let Φ : r r X 99K P be a rational map with its image variety Y ⊂ P . Denoting k(X) (resp. k(Y )) the field of fraction of X (resp. Y ), Φ defines a morphism Φ∗ : k(Y ) → k(X) and Φ is birational if Φ∗ is an isomorphism. This is equivalent to the fact there exist a rational map Ψ : Y 99K X such that the compositions Φ ◦ Ψ = idY and Ψ ◦ Φ = idX as rational maps. 1.3. RATIONAL MAPS 27

1.3.2 Construction with homogeneous polynomials Our main source of examples and applications come from the situation where n n X = P and r = n. In this case, every line bundle L is isomorphic to OP (δ) for 0 n n n an integer δ where OP (δ) is a line bundle with the property that H P , OP (δ) is isomorphic to the space k[x0, ··· , xn]δ of homogeneous polynomials in n + 1 0 n variables of degree δ. Hence we can identify the sections φi ∈ H (L, OP (δ)) with homogeneous polynomials of degree δ. Now, letting V be an n+1 dimensional sub- 0 n r n space of H (L, OP (δ)), a rational map Φ between two projective spaces P and P n n over a field k, denoted Φ : P 99K P , is the data of n+1 homogeneous polynomials φ0, . . . , φn ∈ k[x0, ··· , xn] of the same degree δ without common component. n n n n Given Φ : P1 99K P2 and another rational map Ψ : P2 99K P3 defined by n + 1 homogeneous polynomials ψ0, . . . , ψn ∈ k[y0, . . . , yn] such that Im(Φ) is not contained in the base locus of Ψ (this is the case for instance if Φ is dominant), n n the composition Ψ ◦ Φ of Φ and Ψ is the map from P1 to P3 defined by the polynomial ψ0(φ0, . . . , φn), . . . , ψn(φ0, . . . , φn) where we substitute the variables yi by the polynomials φi. We saw in Example1 that there a preferred representative of rational map of Pn, this motivates the following comment. n n Remark 1.3.5. Given a rational map Φ : P 99K P , we may always consider that the map is defined by polynomials φ0, . . . , φn without common factor. This is requiring that the codimension of the base locus Z = {φ0 = ... = φn = 0} is greater than 1.

1.3.3 Multidegree of a rational map and cycles in a Segre product

We present first the topological degree. Let Φ : X 99K Y be a dominant rational map between two irreducible varieties X and Y of the same dimension over an algebraically closed field k. This situation corresponds to a field extension Φ∗ : k(Y ) → k(X) between the respective fraction fields of X and Y . Definition 1.3.6. The topological degree of Φ is the degree

dt(Φ) = [k(X) : k(Y )] of the ﬁeld extension of k(X) over k(Y ). From Deﬁnition 1.3.4, we have the following relation between topological degree and rationality.

Proposition 1.3.7. The map Φ is birational if and only if dt(Φ) = 1.

Given that Φ : X 99K Y is dominant between two varieties of the same dimension over an algebraically closed ﬁeld k, we explain now a geometric interpretation of the topological degree following [Har92, 7.16].

Proposition 1.3.8. The topological degree dt(Φ) of Φ is equal to the number of points in a general ﬁbre of Φ, i.e. the inverse transform of a general point of Y under Φ. 28 CHAPTER 1. DEGREES OF RATIONAL MAPS

Proof. First we can assume that X and Y are aﬃne open subsets since the content of the proposition is local. So we can assume that X ⊂ An and Y ⊂ Am and Φ is the projection from the graph Γ ⊂ An × Am of Φ on Y which is the restriction of a linear projection An × Am → Am to Γ. It is thus enough to prove the claim for a map Φ : X 99K Y of aﬃne varieties given as the restriction of the projection

p : n n−1 A A (x1, . . . , xn) (x1, . . . , xn−1)

In this case the fraction ﬁeld k(X) of X is generated over k(Y ) by the element xn. Since X and Y have the same dimension and Φ is dominant, xn is algebraic over k(Y ). Thus, let ai ∈ k(Y ) for i ∈ {0, . . . , d} such that

d d−1 G(x1, . . . , xn) = a0(x1, . . . , xn−1)xn + a1(x1, . . . , xn−1)xn + ... is the minimal polynomial satisﬁed by xn (so [k(X) : k(Y )] = d). After clearing denominators, we may take the ai to be regular functions on Y i.e. polynomials in x1, . . . , xn−1. Let ∆(x1, . . . , xn−1) be the discriminant of G as a polynomial in xn. Since G is reduced in k(Y )[xn] and k is algebraically closed, ∆ cannot vanish identically on Y . It follows that the loci {a0 = 0} and {∆ = 0} are proper subvarieties of Y and, on the complement of their union, the ﬁbres of Φ consist of exactly d points.

Example 1.3.9. To illustrate Proposition 1.3.8 we explain the following compu- n n tational way to determine the topological degree of a rational map Φ : P1 99K P2 given by n + 1 homogeneous polynomials φ0, . . . , φn in n + 1 variables each one of n n degree δ and such that dim V(φ0, . . . , φn) ≤ n − 2. Here we let P1 and P2 be two projective spaces of dimension n over k with respective coordinates x0, . . . , xn and y0, . . . , yn.

n (1) The ﬁrst step is to take a general point in P2 . By B´ezout’stheorem, this is n the same as the choice of n general hyperplanes H1,...,Hn in P2 intersecting precisely at this point. In the following, we let:

Hi : ai0y0 + ... + ainyn = 0

be the equations of the hyperplanes Hi. Computationally, one often works with a finite fields or even prime fields Z/pZ. Under many points of view, this is a good approximation of infinite fields and even of algebraically closed fields if the prime number p is big enough. Over finite fields, ”random” means that all elements of the field have the same probability of being chosen.

(2) It remains to compute the ﬁbre Fy of the point y in the intersection ∩iHi. n Let us remark that the ﬁbre {x ∈ P1 , Φ(x) = y} of y is contained in the ∗ ∗ n intersection ∩iΦ Hi where Φ Hi are hypersurfaces of P1 given by the equations: ∗ Φ Hi : ai0φ0 + ... + ainφn = 0. 1.3. RATIONAL MAPS 29

n Actually, if for any x ∈ P1 there exists i ∈ {0, . . . , n} such that φi(x) 6= 0 then x is in the ﬁbre of y. This is just saying that Φ(x) is well deﬁned (x is not in the base base locus of Φ) and that y is the only point in the intersection ∩iHi.

∗ (3) The ﬁbre of y is in general strictly contained in ∩iΦ Hi since the latter scheme contains also the base locus Z = V(φ0, . . . , φn) of Φ. The computational step ∗ consisting in removing Z from ∩iΦ Hi is the saturation

∞ i I y = [IH : IZ ] = ∪ [IH : IZ ] F i≥1

∗ where IH is the ideal of ∩iΦ Hi, IZ is the ideal of Z and

i i [IH : IZ ] = {r ∈ k[x0, . . . , xn], ∃f ∈ IZ , rf ∈ IH }

i is the ideal quotient of IH by IZ [Eis95, Page 15]. Indeed the saturation of an ideal I with respect to an ideal J corresponds to consider all the primary components of I which are not contained in any inﬁnitesimal neighbourhood of V(J). We refer to [CLO07, 4.7 Primary Decomposition of Ideals] for the deﬁnition of primary components of an ideal and [CLO07, 4.4 Zariski Closure and Quotients of Ideals, Exercices 8 to 10] for the geometric meaning of the saturation and its actual computation via Groebner basis algorithms. The

ideal IFy is then the ideal of the ﬁbre Fy.

Now, we turn to the more general notion of multidegree of a rational map. Let X be a smooth irreducible subvariety of Pn, Y be a smooth irreducible subvariety m of P and Φ : X 99K Y be a rational map from X to Y . We denote by Γ the graph of Φ. The situation is summed up with the following commutative diagram:

Pn × Pm

p1 p2 Γ σ1 σ2

n m P Φ P

where p1 (resp. p2) is the ﬁrst (resp. second) projection.

Proposition 1.3.10. [Dol00, 7.1.3] The Chow group of Pn × Pm is the tensor product of the Chow group of Pn with the Chow group of Pm.

n m i n j m ⊕ CH (P ) ⊗ CH (P ) ' ⊕ ⊕ Z ⊗ Z, i,j≥0 i=0 j=0

∗ n m Given i ∈ {1, 2}, we denote by hi the cycle class of pi (Hi) in CH(P × P ) n m where H1 (resp. H2) is the class of a hyperplane in P (resp. P ). Each cycle of n m n m P P i j P × P decomposes into a sum dijh1h2. i=0j=0 30 CHAPTER 1. DEGREES OF RATIONAL MAPS

Definition 1.3.11. The multidegree of a subvariety G of dimension k in Pn ×Pm is k P i k−i the sequence of coefficients (d0, . . . , dk) in the decomposition of [G] = dih1h2 . i=0 n Definition 1.3.12. The multidegree of a rational map Φ : X 99K Y with X ⊂ P and Y ⊂ Pm is the multidegree of the graph Γ of Φ in Pn × Pm. Since dim(Γ) = dim(X), we decompose [Γ] as follows:

k X i k−i [Γ] = dih1h2 . i=0

th The coefficient di is called the i projective degree of Φ . Remark 1.3.13. As defined in the generality of Definition 1.3.12, the nth projective degree dn(Φ) of a rational map Φ does not recover completely the first definition of topological as degree of the general fibre and we illustrate why in the following example. Let C be the image in k3 of the map

φ : k k3 2 3 x0 (x0, x0, x0)

C is actually a smooth rational cubic curve usually called a twisted cubic. In homogeneous coordinates, C is the image of the map

Φ: P1 P3 3 2 2 3 (x0 : x1) (x0 : x0x1 : x0x1 : x1)

This latter map is birational onto its image C so dt(Φ) = 1 with respect to Definition 1.3.6. However the cycle class of the graph ΓΦ of Φ in the Chow ring of P1 × P3 is: 3 2 3 [ΓΦ] = (h1 + h2) = 3h1h2 + h2 ∗ ∗ where h1 = p1H1 (resp. h2 = p2H1) is the cycle class of the pull back of a 1 3 1 3 1 1 3 3 hyperplane of P (resp. P ) and where p1 : P × P → P (resp. p2 : P × P → P ) 3 is the first projection (resp. second). Hence d1(Φ) = 3, the degree of C in P . For more precision about this justification and the cycle class of a variety in P1 × P3 we refer to Section 1.2. Our justification to maintain Definition 1.3.12 is that our results concern dom- n th inant rational maps Φ : X 99K P where the definition of n projective degree and topological degree coincide. More generally, with the previous notation, suppose that Φ is dominant over Y and that dim(X) = dim(Y ) = k. Then

Proposition 1.3.14. Let Φ: X 99K Y be a rational map and let ﬁx the embedding X ⊂ Pn and Y ⊂ Pm. Then

dk(Φ) = deg(Y )dt(Φ). 1.4. THE CREMONA GROUP 31

th i.e. the topological degree dt(Φ) of Φ is the k projective degree of Φ divided by the degree of Y in Pm. Remark 1.3.15. With the notation as in Deﬁnition 1.3.12, let i ∈ {0, . . . , k} and i n k−i let H1 be a general (n − i)-plane of P1 , H2 be a general (m − k + 1)-plane of m i −1 k−i P2 . Then if the intersection H1 ∩ Φ (H2 ) is reduced and 0-dimensional, the th i projective degree di(Φ) of Φ is:

i −1 k−i di(Φ) = card H1 ∩ Φ (H2 )

0 n 0 m where, by convention, we set H1 = P and H2 = P .

1.4 The Cremona group

n n Deﬁnition 1.4.1. Given an integer n ≥ 1, a rational map Φ : P 99K P of topological degree dn(Φ) = 1 is called a Cremona map.

Deﬁnition 1.4.2. Given n ≥ 1, the set of all Cremona maps of Pn has a structure of group with respect to the composition law of rational maps. We call this group the Cremona group of Pn, denoted by Bir(Pn).

1.4.1 First properties of the Cremona group There are a lot of questions and results about the Cremona group. We restrict our presentation to the foundational result of Noether and Castelnuovo establishing that the Cremona group Bir( 2 ) over the complex field has a remarkable set of PC generators. We define first the projective automorphisms of Pn.

Deﬁnition 1.4.3. Let n ≥ 1, to any matrix A = (aij)0≤i≤n,0≤j≤n ∈ PGln+1, we n n P can associate a birational map sending (x0 : ... : xn) ∈ P to ( a0jxj : ... : j=0 n P −1 anjxj), the inverse being the map associated to A . Such a map is called a j=0 projective automorphisms of Pn. Theorem 1.4.4. (Castelnuovo-Noether’s theorem) The Cremona group Bir( 2 ) is PC generated by the projective automorphisms of P2 and the standard Cremona map τ = (x1x2 : x0x2 : x0x1) We refer to [Dol11, 7.5] for a proof of this result.

Example 1.4.5. Let

Φ: P2 P2 2 2 (x0 : x1 : x2) (x0 : x0x1 : x1 − x0x2).

It is an involution, i.e. Φ ◦ Φ = id and we can compute that

Φ = A4 ◦ τ ◦ A3 ◦ τ ◦ A2 ◦ τ ◦ A1 32 CHAPTER 1. DEGREES OF RATIONAL MAPS

0 1 0  4 0 1 1 1 0 where A4 = 0 1 −2, A3 = 1 1 0, A2 = 0 −1 0, 1 1 0 0 1 0 0 0 1  1 1  2 2 0 1 1 A1 =  2 − 2 0 and τ = (x1x2 : x0x2 : x0x1). 0 0 1

An output of Theorem 1.4.4 is that Bir( 2 ) has a rather simple set of generators. PC This no longer true when n ≥ 3 as stated by the following result.

Theorem 1.4.6. [Pan99] When n ≥ 3, the Cremona group Bir( n) cannot be PC generated by a set of Cremona maps of bounded degree.

Determinantal rational maps Let n ≥ 1 and M be a matrix of size (n + 1) × n whose entries are homogeneous polynomials in k[x0, ··· , xn] and such that all the entries of one given column have the same algebraic degree. Then the n × n-minors of M deﬁne a rational map of Pn.

Deﬁnition 1.4.7. Given n ≥ 1 and M ∈ Matn+1,n(k[x0, ··· , xn]) with homogeneous entries, a rational map deﬁned by the n × n-minors of M is called a determinantal rational map. If this map is birational we call it a determinantal Cremona map.

Example 1.4.8. Over k[x0, x1, x2], let Φ be the determinantal rational map de- ﬁned by the 2 × 2-minors of   x0 x1x2 2 M = x1 x0  . 0 x0x2

That is, the map Φ is represented by the polynomials

3 2 2 (x0 − x1x2, x0x2, x1x0x2) and is actually a Cremona map of P2 whose base locus Z is supported over the points (0 : 0 : 1) and (0 : 1 : 0). Its inverse Φ−1 is deﬁned by the polynomials

2 2 3 3 (−x0x1 − x1x2, −x0x1x2 − x2, −x1) which are the 2 × 2-minors of the matrix

 2  −x2 −x1 0 M =  x1 0  . 2 0 x0x1 + x2

In P2, it is remarkable that all Cremona maps of algebraic degree 2 are actually determinantal. Such a result follows from the classical description of Cremona map of algebraic degree 2 (see [Dés12]). When looking at a list of Cremona maps of algebraic degree 3 such as in [Dés12, 4.6], we compute that the first members 1.4. THE CREMONA GROUP 33 of the list are also determinantal. Since the standard Cremona map τ is itself   x0 x0 determinantal as the 2×2-minors of the matrix  0 x1 and since τ is the only −x2 0 generator of Bir(P2) of algebraic degree not equal to 1, we end up with ProblemE. The answer is negative as illustrated by the following map Φ in Proposition 1.4.9 which is the composition the map defined by the 2 × 2-minors of the matrix   x1x2 x0x2 x0x2 x0x1 x0x1 x1x2 and the standard Cremona map τ = (x1x2, x0x2, x0x1).

2 2 Proposition 1.4.9. The map Φ: P 99K P defined by the polynomials  φ = x3x2 − x x3x − x2x x2 + x2x3  0 0 1 0 1 2 0 1 2 1 2 2 3 3 2 2 2 3 φ1 = −x0x1 + x0x1x2 + x0x1x2 − x0x2  2 2 3 2 3 2 3 φ2 = −x0x1x2 + x0x2 + x1x2 − x0x1x2 is a Cremona map of P2 which is not determinantal. Proof. We introduce several tools that we will use in the next chapters. Over the 2 coordinate ring R = k[x0, x1, x2] of P , we let IZ = (φ0, φ1, φ2). Of course the base ideal sheaf IZ of Φ defined in the first section is just the sheafification of IZ . Now consider a minimal free presentation of IZ

M 3 F R IZ 0 (1.4.1) where F is a free R-module and M is the presentation matrix of IZ . Since the base locus Z of Φ is assumed to be of codimension strictly greater than 1, the Hilbert- Burch theorem [Eis95, 20.15] states precisely that rank(F ) = 2 if and only if IZ is generated by the 2 × 2-minors of M. But in our case, a minimal free presentation of IZ actually reads

3 M 3 R R IZ 0 (1.4.2) and  2 2 2 2  x0x1 − x0x2 x1x2 − x0x2 0 2 2 2 2 M = x0x1 − x1x2 0 −x0x2 + x1x2 . 2 2 2 2 0 x0x1 − x1x2 x0x1 − x0x2 Hence Φ is not determinantal and it is a computation, for example using the algorithm we explained in Example 1.3.9 that Φ is birational. i1 : k = QQ o1 = QQ o1 : Ring 34 CHAPTER 1. DEGREES OF RATIONAL MAPS

i2 : R = k[x_0..x_2] o2 = R o2 : PolynomialRing i3 : I = ideal(x_0^3*x_1^2-x_0*x_1^3*x_2-x_0^2*x_1*x_2^2+x_1^2*x_2^3, -x_0^2*x_1^3+x_0^3*x_1*x_2+x_0*x_1^2*x_2^2-x_0^2*x_2^3, -x_0^2*x_1^2*x_2+x_0^3*x_2^2+x_1^3*x_2^2-x_0*x_1*x_2^3); o3 : Ideal of R i4 : J = ideal ( (gens I)*random(R^{3:1},R^{2:1}) ); o4 : Ideal of R i5 : degree saturate(J,I) o5 = 1 hence the ﬁbre of a general point has cardinal 1 so the topological degree of Φ is 1.

Remark 1.4.10. Let us explain why we restrict ProblemE to birational maps of P2 of degree stricly greater than 1. It is because the minimal free resolution of the 2 base ideal IZ of any projective automorphism of P reads

3 3 0 R R R IZ 0.

Indeed, Z beeing empty, the three generators of IZ are a regular sequence (see Deﬁnition 2.2.2) so IZ is resolved by the Koszul complex associated to the three 2 generators of IZ (see Deﬁnition 2.2.5). Hence any projective automorphism of P is not determinantal. Part I

Projectivization of an ideal sheaf

Chapter 2

Torsion of the symmetric algebra

In this chapter, we consider a smooth quasi-projective variety X. Recall from n Subsection 1.3.1 that a rational map Φ : X 99K P defines an ideal sheaf IZ which is the image of the evaluation morphism V ⊗OX → L, where L is a line bundle over X and V is the subspace of global sections of L defining Φ. The main idea here is to consider the projectivization P(IZ ) of IZ . A locally n free presentation of IZ determines an embedding of P(IZ ) in PX . Since P(IZ ) ˜ contains also the blow-up X of X along Z = V(IZ ), the birationality of Φ and, more generally, the multidegree of Φ, defined over X˜, can be studied in terms of the properties of P(IZ ) (see Subsection 1.3.3 for the definition of the multidegree). A locally free presentation of an ideal sheaf IZ generated by n + 1 global sections (φ0, . . . , φn) of a line bundle L is the data of an exact sequence:

M φ0 . . . φn F V ⊗OX IZ ⊗ L 0

where F is locally free. The matrix M is called a presentation matrix of IZ . Since n n our applications concern mostly rational maps Φ : P1 99K P2 , we emphasize that, in this case, we identify the sections φi with their corresponding homogeneous n polynomials in R = k[x0, ··· , xn] (with P1 = Proj(R)) of the same degree δ and that the ideal sheaf IZ is then the sheafification of the ideal IZ generated by polynomials φ0, . . . , φn over R. With this identification a presentation of IZ is then the sheafification of a presentation of IZ φ0 . . . φn M n+1 F R IZ 0 where F is a free R-module. In this perspective, we can apply results from commutative algebra and eventually make the computation with a computer algebra system such as Macaulay2 to compute the free presentation of IZ and to infer a presentation of IZ . As a matter of notation, we denote by M both a matrix presentation of IZ and the sheafified map M˜ of M. The justification for this sim- plification is that the entries of M can be identified with their corresponding global M sections which determine the map F → V ⊗O n where F is the sheafification of F . P1

37 38 CHAPTER 2. TORSION OF THE SYMMETRIC ALGEBRA

2.1 Projectivization and blow-up

In the following, we will use that the formation of the symmetric algebra commutes with base change. Let us explain now what we mean by that.

Proposition 2.1.1. [Eis95, Proposition A.2.2] Let R be a ring, R0 be an R-algebra and M be an R-module. Then there is a R-module isomorphism

0 0 Sym(M ⊗R R ) ' Sym(M) ⊗R R .

In our context, from a relative point of view, this implies that the localization of the symmetric algebra of an ideal sheaf IZ is the symmetric algebra of the localization of IZ . Notation 2.1.2. Letting X and G be as in Deﬁnition 1.1.3 and given a subscheme L of P(G) and any x ∈ X, we denote by Lx the scheme-theoretic ﬁbre of the projection π : P(G) → X restricted to L above x.

2.1.1 Projectivization of an ideal sheaf We consider now the special case of the projectivization of an ideal sheaf. Let X be a smooth quasi-projective variety, let I be an ideal sheaf over X and let

r+1 (φ0 ... φr ) OX I ⊗ L 0, (2.1.1) be a surjection for some ample line bundle L on X. We have an embedding of r r+1 P(I ⊗ L) in PX . Indeed, by [Bou70, A.III.69.4], the surjection OX → I ⊗ L → 0 r+1 of (2.1.1) implies that Sym(OX ) surjects over Sym(I ⊗ L) and that the kernel of r+1 this latter surjection is generated by the kernel of OX → I ⊗L. Hence denoting F r+1 the kernel of (φ0 . . . φr) in (2.1.1) and M the morphism F → OX , we have that the ideal sheaf IP(I) of P(I) is locally generated by the entries of the row matrix ∗ y0 . . . yr p M where y0, . . . , yr are the relative homogeneous coordinates of r r PX and p : PX → X is the projection. Now by Lemma 1.1.2, we have that P(IZ ) ' P(IZ ⊗ L) and moreover that:

O (1) 'O (1) ⊗ p∗L∨. P(IZ ) P(IZ ⊗L)

r So we consider that P(IZ ) itself is embedded in PX . Let us summarize these properties with the following proposition.

Proposition 2.1.3. Let X be a smooth quasi-projective variety, L be an ample line bundle on X and let I be an ideal sheaf of X. A locally free presentation of I ⊗ L, M r+1 (φ0 ... φr ) F OX I ⊗ L 0 (2.1.2)

r determines a closed embedding of the projectivization X of IZ into PX . The r ideal sheaf IX of X in PX is locally generated by the entries of the row matrix ∗ y0 . . . yr p M where y0, . . . , yr are the relative homogeneous coordinates of r r PX and p : PX → X is the projection. 2.1. PROJECTIVIZATION AND BLOW-UP 39

Remark 2.1.4. In the following, except in the examples, we consider locally free presentation m ∨ M r+1 ∨ (φ0 ... φr ) OX ⊗ L OX ⊗ L I 0 (2.1.3)

r+1 ∨ of I. In this case, the projectivization P(I) is embedded in P(OX ⊗ L ). As r we saw in Lemma 1.1.2, this is still considering P(I) in PX but with a modiﬁed hyperplane class. This makes the redaction and veriﬁcation of the proves easier.

We illustrate the previous notions with the following situation where the base ﬁeld k is any ﬁeld. 2 2 3 Let IC = (x0x2 − x1, x0x3 − x1x2, x1x3 − x2) be the ideal sheaf over P given by the 2 × 2 minors of the matrix

x x x M = 0 1 2 x1 x2 x3 and denote those three minors by φ0, φ1, φ2. As we explained in Remark 1.3.13, 3 the subscheme C = V(IC) of P is a smooth rational curve called the twisted cubic. We summarize this situation into the following exact sequence:

t 2 M 3 (φ0 φ1 φ2) 0 O 3 (−1) O 3 IC(2) 0 (2.1.4) P P Now, denoting by X the projectivization P IC(2) of IC(2), (2.1.4) determines 3 3 3 2 a closed embedding of into (O 3 ). Moreover, denoting (O 3 ) by × with X P P P P P P variables xi (resp. yi) for the ﬁrst (resp. second) factor, the ideal of IX of X in P3 × P2 is generated by the entries in the row matrix

t (y0 y1 y2) M = y0x0 + y1x1 + y2x2 y0x1 + y1x2 + y2x3

t 3 since by deﬁnition Im( M) is the kernel of O 3 → IC(2) → 0. P By computing the primary decomposition of IX, for example using basic functions of Macaulay2, we deduce that X has codimension 2 in P3 × P2 and that it is irreducible. Now, we study the ﬁbres of the morphism π : X → P3.

(i) If z∈ / C, then the localization IC,z of IC at z is equal to Oz. Since the formation of the symmetric algebra commutes with base change (Proposi- tion 2.1.1), the ﬁbre Xz is isomorphic to {z}. Actually, over the open subset 3 U = P \C, we show by the same argument that XU ' U since IU 'OU . 1 0 t (ii) Now let z = (1 : 0 : 0 : 0) so Mz = 0 0. 0 0 Hence, since the formation of the symmetric algebra commutes with base t change and since X has equation y0 y1 y2 M, the ﬁbre Xz of z in X is 2 a line in Pz.

As we are going to see in Subsection 2.1.2, X corresponds exactly here to the blow-up of P3 along C. 40 CHAPTER 2. TORSION OF THE SYMMETRIC ALGEBRA

Another perspective We emphasize the following point. Since I is generated by the 2 × 2-minors of the C x0 x1 x2 matrix M = , the structure sheaf OC of C ﬁts also in the following x1 x2 x3 exact sequence:

 2  x2 − x1x3 −x1x2 + x0x3 2 x1 − x0x2 3 M 2 0 O 3 (−3) O 3 (−1) O 3 OC 0 P P P

2 3 1 In this case, = OC is embedded in (O 3 ) ' × and is a complete D P P P P P 3 1 3 1 intersection of three divisors of class c1 OP ×P (0, 1) in the Picard group of P ×P . Moreover, writing down explicitly those three divisors, we see that the ﬁbre Dx of D over a point x ∈ P3 is deﬁned by the ideal

(y0x0 + y1x1, y0x1 + y1x2, y0x2 + y1x3) thus:

(1) if x ∈ C (i.e. rank(Mx) = 1), dim(Dx) = 0,

(2) whereas if x∈ / C (i.e. rank(Mx) = 2), Dx is empty. Hence π(D) is set theoretically equal to C. Actually, by deﬁnition of the structure sheaf of the projectivization P(OC) of OC we have also that π∗(OD) = OC.

2.1.2 Blow-up of an ideal sheaf We turn now to the study of the Rees algebra of an ideal sheaf following [Har77, II.7]. Here, we explain how the data of the Rees algebra of the ideal sheaf of the base locus of a rational map Φ is equivalent to the data of the graph Γ of Φ. For the rest of the subsection, X is a smooth quasi-projective variety. Deﬁnition 2.1.5. Let I be a coherent ideal sheaf on X. Consider the Rees algebra

d d R(I) = ⊕d≥0I t ⊂ OX [t]

d th 0 where I is the d power of the ideal I, and where we set I = OX . Then R(I) satisﬁes( 4). We call Proj R(I) the blow-up of X with respect to the coherent sheaf of ideal I and we denote it by X˜. If Z is the closed subscheme of X deﬁned by I, then we also call X˜ the blow-up of X along Z.

Example 2.1.6. Let X be the projective space P3 over any ﬁeld k and let I be the ideal sheaf associated to the ideal (x0, x1, x2). We denote also by z the point 3 d V(x0, x1, x2) in P . In this case the component I of degree d of R(I) is generated d d by all the monomials in x0, x1, x2 of degree d. In other words R(I) = ⊕d≥0I t ˜3 is isomorphic to k[x0, x1, x2] and the blow-up P with respect to I is equal to X = P(I). So, as in Section 1.1, let: 2.1. PROJECTIVIZATION AND BLOW-UP 41

  0 x0 −x1 −x0 0 x2  x1 −x2 0 x0 x1 x2 3 3 O 3 (−1) O 3 I(1) 0 P P

be a locally free presentation of I(1). By Lemma 1.1.2, P I(1) is isomorphic to 3 3 2 so the ideal I of in (O 3 ) ' × is equal to X X X P P P P

(y0x1 − y2x0, −y0x2 + y2x0, y1x2 − y2x1).

We deduce from those equations, for example by computing its primary decom- ˜3 position with Macaulay2, that P is irreducible of dimension 3. Furthermore, 3 ˜ −1 letting U = P \{z}, we have that XU ' U and that σ ({z}) is isomorphic to 2 2 Pz '{z} × P . d The fact that S = ⊕d≥0I is a sheaf of integral domains on X explains Item (a) of the following proposition. We refer to [Har77, II.7.16] for the proof of Item (b) and Item (c).

Proposition 2.1.7. Let I ⊂ OX be a non zero coherent sheaf of ideals on X, and let σ : X˜ → X be the blowing-up with respect to I. Then: (a) X˜ is a variety, (b) σ is a birational, proper, surjective morphism, (c) if X is quasi-projective over k then X˜ is also, and σ is a projective morphism.

Now, let I be the base ideal sheaf of a rational map Φ = (φ0 : ... : φn): X 99K 0 Pn associated to an n + 1-subspace V of H (X, L) where L is a line bundle over X as in Section 1.3. We denote by Z the base scheme V(I) in X. Proposition 2.1.8. The blow-up σ : X˜ → X along Z is isomorphic to the graph n Γ of Φ embedded into PX .

Proof. Let U = W \Z. The representative hΦU ,Ui of Φ to U is a morphism. Hence n the graph ΓΦU ' U of ΦU is integral and its closure Γ in PX is also integral. Moreover Γ is a priori a subscheme of X˜. This follows from the universal property of X˜ [Har77, II.7.14]. Hence, since X˜ is also integral and coincide with Γ over U, by Proposition 2.1.7 (a), Γ and X˜ coincide.

Example 2.1.9. Let X be a smooth quasi-projective variety over k and let I = (φ0, . . . , φn) be an ideal sheaf given by n + 1 global sections of a line bundle L over n X. We denote by Φ : X 99K P the associated map. Proposition 2.1.8 gives a ﬁrst approximation to compute the equations of the blow-up X˜ of X with respect to I. Indeed, in X ×Pn with variables y in the second φ0 . . . φn factor, let K be the scheme given by the 2×2 minors of the matrix . y0 . . . yn Since X˜ is isomorphic to the graph Γ of Φ by 2.1.8, we have that K contains X˜. As we will see, K contains also the projectivization X of I. 42 CHAPTER 2. TORSION OF THE SYMMETRIC ALGEBRA

2.2 Symmetric algebra versus Rees algebra

We explain now the relation between the blow-up of a given (coherent) ideal sheaf I on a smooth quasi-projective variety X and the projectivization P(I) i.e. the relation between the Rees algebra of I and the symmetric algebra of I. As in the previous section, X is a smooth quasi-projective variety, I is a coherent ideal sheaf over X and we denote by X˜ the blow-up of X along I and by X the projectivization of I.

Proposition 2.2.1. The blow-up X˜ is an irreducible component of X. Proof. Denoting T (I) the tensor algebra associated to I and T n(I) its component of degree n, let Qn : T n(I) → Intn be the surjection sending a pure tensor n n x1 ⊗ ... ⊗ xn ∈ T (I) to x1 . . . xnt . It induces a surjection Q : T (I) → R(I) whose kernel contains the symmetric relations, i.e. the subsheaf generated by all the sums x ⊗ y − y ⊗ x. Hence Q induces a surjection R : Sym(I) → R(I). This shows the injection X,˜ → X at the Proj level. Since X˜ is integral by 2.1.7 (a) and coincide with X over the complement of V(I) in X, X˜ is an irreducible component of X.

Of course, the blow-up X˜ and the projectivization X are equal when the surjection R : Sym(I) → R(I) is injective and we explain now a suﬃcient condition when it is the case.

2.2.1 Local complete intersections We define first regular sequences in the local case. Recall that given a module M over a ring R, an element r ∈ R is called a nonzerodivisor if rm 6= 0 for all m ∈ M\{0}. Definition 2.2.2. Let R be a noetherian ring and M an R-module. A sequence of elements r0, . . . , rn ∈ R is called M-regular if:

(1)( r0, . . . , rn)M 6= M,

(2) for i = 1, . . . , n, ri is a nonzerodivisor on M/(r1, . . . , ri−1)M. When M = R, a R-regular sequence is simply called a regular sequence. Example 2.2.3. With R = k[x0, x1, x2], the sequence x0, (1 − x0)x1, (1 − x0)x2 is regular whereas (1 − x0)x1, (1 − x0)x2, x0 is not. Deﬁnition 2.2.4. Let R be a noetherian ring, M an R-module and I an ideal of R. If IM 6= M, then by [Eis95, Theorem 17.4], the length of all maximal M- regular sequences in I are the same. We deﬁne the depth of I on M, denoted by depth(I,M), to be the length of any maximal M-regular sequence in I. If M = R, we call just the depth of I, written depth(I).

One interest in regular sequences (r0, . . . , rn) lies in the remarkable structure of the resolution of the module M = R/I where I is the ideal generated by r0, . . . , rn. Namely the Koszul complex resolves I. We introduce the Koszul complex following [Har77, III.7]. 2.2. SYMMETRIC ALGEBRA VERSUS REES ALGEBRA 43

Definition 2.2.5. Let R be a ring and r0, . . . , rn ∈ R. We define the Koszul complex K•(r0, . . . , rn) as follows: K1 is a free R-module of rank n + 1 with basis p e0, . . . , en. For each p = 0, . . . , n, Kp = ∧ K1. We define the boundary map d : Kp → Kp−1 on the basis vectors:

p X j−1 d(ei1 ∧ ... ∧ eip ) = (−1) rij ei1 ∧ ... ∧ eˆij ∧ ... ∧ eip j=1 which veriﬁes that dp ◦ dp−1 = 0 i.e. K•(r0, . . . , rn) is a complex of R-module. If M is any R-modules, we set K•(r0, . . . , rn,M) = K•(r0, . . . , rn) ⊗ M. The homology of the complex of R-modules K•(r0, . . . , rn,M) is called the Koszul homology associated to the Koszul complex.

Proposition 2.2.6. [Har77, Proposition III.7.10A] Let R be a noetherian ring, M an R-module and r0, . . . , rn a M-regular sequence, then: Hi K•(r0, . . . , rn,M) = 0 for i > 0 and H0 K•(r0, . . . , rn,M) = M/(r0, . . . , rn)M

th where Hi K•(r0, . . . , rn,M) is the i homology module of the complex of R- modules K•(r0, . . . , rn,M).

Example 2.2.7. Let R = k[x, y, z] and I be the ideal (x, y, z). The sequence (x, y, z) is regular so the Koszul complex K•(x, y, z, R) is a free resolution of R/I. Namely:

x  0 z −y y −z 0 x  z y −x 0 x y z 0 R R3 R3 I 0 is an exact complex.

We explain now how the notion of regular sequence is relevant when considering Rees and symmetric algebra.

Proposition 2.2.8. If I = (x0, . . . , xn) ⊂ R is generated by a regular sequence, then R(I) ' Sym(I) ' R[y0, . . . , yn]/J where J is generated by the 2 × 2 minors of the matrix

x . . . x 0 n . y0 . . . yn

Proof. Since (x0, . . . , xn) is regular, the Koszul complex K•(x0, . . . , xn,R) resolves R/I so I has the following presentation: 44 CHAPTER 2. TORSION OF THE SYMMETRIC ALGEBRA

  −x1 x2 0      x0 0     0 −x   0     0 0     x   n   0 0 −x  n−1 x0 . . . xn n+1 n+1 R( 2 ) R I 0 where the presentation matrix is the second diﬀerential D2 of the Koszul complex. Actually, the ideal generated by the row matrix (y0 . . . yn)D2 is the same as the x0 . . . xn one generated by the 2 × 2 minors of the matrix , D2 being the y0 . . . yn presentation matrix of the ideal (x0, . . . , xn). Hence Sym(I) ' R[y0, . . . , yn]/J. The proposition follows from the isomorphism Sym(I) 'R(I) implied by the fact that a permutation in a regular sequence is a regular sequence (see [Eis95, 17.2] and [EH00, Exercise IV-26] for a complete treatment of this last argument).

Example 2.2.9. In the case of an ideal I = (x, y) generated by a regular sequence of length 2, we thus have

R(I) ' Sym(I) ' R[u, v]/(uy − vx).

As an illustration of Proposition 2.2.8, assume that I = (x0, . . . , xn) is generated by n + 1 elements of R and let

Rm M Rn+1 I 0 be a presentation of I. The generators of the symmetric algebra Sym(I) of I viewed as a quotient of R[y0, . . . , yn] are the entries in the row matrix (y0 . . . yn)M as we explained in Section 1.1. Hence if R(I) = Sym(I), the generators of R(I) are linear in the y variables. This motivates the following deﬁnition.

Deﬁnition 2.2.10. [Vas05] Given a noetherian ring R and an ideal I over R, I is said of linear type if R(I) = Sym(I).

2.2.2 Torsion of the symmetric algebra Now, we determine the kernel of the surjection Sym(I) → R(I).

Lemma 2.2.11. [Mic64] Let X be a smooth quasi-projective variety and let I be an ideal sheaf over OX . Then R(I) = Sym(I)/T(I) where T(I) is the torsion sheaf {u ∈ Sym(I), ∃r ∈ OX , ru = 0} of the symmetric algebra.

Proof. Since the formation of the symmetric algebra and the Rees algebra commutes with base change (see 2.1.1), we can assume that X is an aﬃne scheme Spec(A) with A a local ring. It is thus enough to show Lemma 2.2.11 for an ideal I generated by φ0, . . . , φn over an integral noetherian ring A. As we explained in 2.2. SYMMETRIC ALGEBRA VERSUS REES ALGEBRA 45

Subsection 2.1.1, by [Bou70, A.III.69.4], Sym(I) is isomorphic to A[y0, . . . , ym]/Q where Q is the A-ideal generated by the linear polynomials b0y0 + ... + bmym such that b0φ0 + ... + bmφm = 0. Consider the surjection

j j g : A[y0, . . . , ym] R(I) = ⊕j≥0I t .

yi φit

First, Q ⊂ Q∞ = ker(g) and, since A is integral, the torsion module T(I) of Sym(I) is also contained in Q∞/Q. Moreover, letting F ∈ A[y0, . . . , ym] and F = F0 + F1 + ... its decomposition into homogeneous polynomials, we have that F ∈ Q∞ if and only if Fi ∈ Q∞ for any i. Moreover, any homogeneous polynomial F is in Q∞ if and only if F (φ0, . . . , φn) = 0. Hence ker Sym(I) → R(I) = Q∞/Q which contains T(I). Now we show the reverse inclusion Q∞/Q ⊂ T(I), that is, for all element f ∈ Q∞ there exist r ∈ A such that rf ∈ Q. We proceed by induction on the degree δ of f. If δ = 1, we have Q∞ = Q by deﬁnition of Q so given any f ∈ Q∞ of degree 1, 1 × f ∈ Q which shows the initialisation of the induction. Now, for any δ > 1, suppose the result true for any P of degree δ − 1 and take f ∈ T(I). Write

f = y0f0(y0, . . . , yn) + y1f1(y1, . . . , yn) + ... + ynfn(yn) and write h = y0f0(φ0, . . . , φn) + y1f1(φ1, . . . , φn) + ... + ynfn(φn). Since h has degree 1 in Q∞ it is in Q. Now write,

δ−1 δ−1 φn f − yn h = y0h0(y0, . . . , yn) + ... + yn−1hn−1(yn−1, yn) for hi homogeneous of degree δ − 1 in Q∞. By the induction hypothesis, there are δ−1 ri ∈ R such that rigi ∈ Q. Then r = r0 . . . rn−1φn ∈ R is an element such that rf ∈ Q.

Geometric description of the torsion We turn to the geometric point of view of Lemma 2.2.11. Let X be a smooth quasi-projective variety and let I be an ideal sheaf on X. We denote by X the ˜ projectivization of I with its structure map π1 : X → X and by X the blow-up of X with respect to I and with structure map σ1 : X˜ → X. The images by π1 of the irreducible components of X different from X˜ are contained in the support of Z. ˜ −1 Indeed, over the set U = X\Z, we have IU = OU , so that X U = X U = π1 (U). This justifies the following definition:

Deﬁnition 2.2.12. An irreducible component of the projectivization X of I d- iﬀerent from X˜ is called a torsion component of X. The union of the torsion components is called the torsion part of X, denoted by TZ .

The following proposition provides a way to detect the torsion components of X and to describe them with algebraic properties of Z. 46 CHAPTER 2. TORSION OF THE SYMMETRIC ALGEBRA

Proposition 2.2.13. Let x ∈ X be a closed point and let

M Φ P2 P1 IZ 0 E (2.2.1) 0 0 be a locally free presentation of IZ where P1 has rank n+1 and where E is the image of M. The scheme-theoretic fibre Xx when equipped with its reduced structure is n−r isomorphic to Px where r = rank(Mx). Proof. Since the formation of the symmetric algebra commutes with base change (see 2.1.1), the fibre Xx is obtained by localizing X at x and taking P(IZ ⊗ kx), where kx is the residue field of OX at x. Now tensor (2.2.1) by kx and observe that the kernel Kx of the surjection Φx : P1 ⊗ kx → IZ ⊗ kx is a quotient of E ⊗ kx, which in turn is a quotient of P2 ⊗ kx. The composition of these surjections and of the inclusion Kx → V is just the matrix Mx, so ker(Φx) = Im(Mx). Therefore dim(IZ ⊗ kx) = n + 1 − rank(Mx), which completes the proof. From a practical point of view, it might be difficult to determine how the fibres (hence, as we will see, the torsion components) vary from a given presentation, as illustrated by the following example:

Example 2.2.14. Consider the ideal I of A = k[x, y, z] generated by the 3 × 3- minors φ0, . . . , φ3 of the matrix

0 xz y2  0 x xy   . x y y  y z x

Since V(I) has the expected codimension 2 in Spec(A), we can apply the Hilbert-Burch theorem (see [Eis95, 20.15] or Proposition 1.4.9 for this theorem) to show that a minimal free resolution reads:

  0 xz y2 0 x xy   x y y  y z x (φ0 . . . φ3) 0 A3 A4 I 0

2 Hence, above the line {x = y = 0} in X = Spec(A), the fibre is {x = y = 0} × Pk 3 but above the point {x = y = z = 0}, the fibre is {x = y = z = 0} × Pk. Actually, after computing the primary decomposition of the projectivization X of I with, for instance, Macaulay2, these fibres are the torsion components of X.

Determinants and Fitting ideals As we saw in the proof of Proposition 2.2.13, the location of the torsion components is computed by the rank of the presentation matrix M motivating the following deﬁnition. 2.2. SYMMETRIC ALGEBRA VERSUS REES ALGEBRA 47

Deﬁnition 2.2.15. Let X be an algebraic variety and F, G be vector bundles on X, of ranks f, g respectively. For any vector bundle map s : F → G and l ∈ N verifying 0 ≤ l ≤ min{f, g}, the l’th degeneracy locus of s is the set

Xl(s) = {x ∈ X| rank s(x) ≤ l}.

Since a linear map has rank ≤ l if and only if all l + 1-minors vanish, the set Xl(s) can also be described as the zero locus of the section ∧l+1s ∈ H0(X, (∧k+1F)∨ ⊗ ∧k+1G). As such, it comes with a natural structure as a closed subscheme of X. We denote by Il(s) its associated ideal sheaf and we call it the ideal sheaf of minors of size l of s by extension. Proposition 2.2.16. [Eis95, 20.4] Let I be an ideal sheaf of the structure sheaf 0 s n+1 0 s OX of a quasi-projective variety X and let F → OX → I → 0 and F → n0+1 0 OX → I → 0 be two presentations of I where F (resp. F ) is locally free of rank r (resp. rank r0). 0 Then, for each integer i ≥ 0, we have that In−i(s) = In0−i(s ). This justifies the following definition. Definition 2.2.17. We define the ith Fitting ideal of I to be the ideal sheaf

Fitti(I) = In−i(s) for a given a presentation morphism s of I whose image has rank r. We describe in more details the special case when I is generated by n + 1 global sections of a line bundle L, n being the dimension of X, and V(I) is zero- dimensional in X. Let (φ . . . φ ) s n+1 0 n F OX I ⊗ L 0 be a locally free presentation of I ⊗ L. The map s can be interpreted dually as the data of n + 1 sections of F ∨.

Proposition 2.2.18. Under the previous settings, the ideal sheaf Fittn−1 I is the ideal sheaf generated by the common vanishing of these n + 1 sections and V(Fittn−1 I) ⊂ Z = V(I).

Proof. Since rank(F) = n, Fittn−1 I is equal to the ideal sheaf I1(M) of the minors 1 × 1 of M which means that it is generated by the common vanishing of the n + 1 ∨ sections of F . The fact that V(Fittn−1 I) ⊂ Z = V(I) follows from [Eis95, 20].

Proposition 2.2.19. Let X be a smooth quasi-projective variety of dimension n over k and let I be an ideal sheaf over X. Denoting Z = V(I), assume that codim(Z) = n and that I ⊗ L is generated by n + 1 sections for some line bundle L over X. Then the images of the torsion components of X in X are precisely supported on the points of the subscheme V(Fittn−1 I). Moreover, each torsion n component is isomorphic to Pk when equipped with its reduced structure. 48 CHAPTER 2. TORSION OF THE SYMMETRIC ALGEBRA

Proof. Since Z is zero-dimensional, any z ∈ Z is in an affine open set U = Spec(A) of X over which L is trivial. So over U, let φ0 . . . φn m M n+1 OU OU I U 0 be a presentation of I. Let z be a (closed) point of V I1(M) i.e. assume that M vanishes at z. Then n the fibre of the projection p : X → X is Pz . This fibre is thus contained in the reduced structure of an irreducible component of X which is distinct from the blow- up X˜ of X with respect to I. It is therefore contained in a torsion component. ˜ This in turn is clear as the restriction of p to X is birational and the p X˜ -fibre at any point has dimension at most n − 1. Hence, we have proved that any point in V I1(M) lies in the image of a torsion component. Conversely, take a (closed) point x ∈ X, put z = p(x) and assume that z lies away from V I1(M) . First, if z∈ / Z then x lies in no torsion component, as all such components map to the base locus Z. Then, if z ∈ Z, since the formation of the symmetric algebra commutes with base change (2.1.1), up to localizing X at z we may assume that X = Spec(A) with A regular local ring. Then, since z∈ / V I1(M) , by [Eis95, Proposition 20.6] we argue that I can be generated (locally around z) by a regular sequence (ψ1, . . . , ψn). Therefore, by Proposition 2.2.8 we have that R(I) = Sym(I) so that there is no torsion component in X. Clearly, this implies that x does not belong to any torsion component. Hence, the torsion part of is supported over n whose irreducible X PV(I1(M)) n components (i.e. the torsion components of X) are supported over Pz for z ∈ V(I1(M)) since Z is 0-dimensional.

Notation 2.2.20. In the situation of Proposition 2.2.19, for every z ∈ Z, we let Tz be the scheme-theoretic ﬁbre of the restriction of π1 to TZ . By Proposition 2.2.19, n n Tz is set-theoretically equal to Pz so Tz = [Tz] · c1 O (1) is a 0-cycle on X. We P X denote by TZ the 0-cycle Tz. z∈Z

Generalised Milnor and Tjurina numbers Anticipating on Subsection 6.2.1 about the possible measures of the difference between Rees and symmetric algebra, let us illustrate here a way to generalise Milnor and Tjurina numbers (see Definition8 and Definition 16 for their usual definitions). So let X be a smooth quasi-projective variety of dimension n, I be an ideal sheaf generated by n + 1 global sections of a line bundle L over X and assume that Z = V(I) is zero-dimensional. Definition 2.2.21. With notation as in Notation 2.2.20, for every z ∈ Z, put:

• τ(Z, z) = length(OZ,z)

• µ(Z, z) = τ(Z, z) + deg(Tz).

We let τ(Z) = P τ(Z, z) and µ(Z) = P µ(Z, z). z∈Z z∈Z 2.2. SYMMETRIC ALGEBRA VERSUS REES ALGEBRA 49

The result is that, when X = Pn, and the n + 1 sections are the partial derivatives of a square free homogeneous polynomial f ∈ k[x0, . . . , xn], the numbers µ(Z) and τ(Z) coincide with the usual Milnor and Tjurina numbers µf (Z) and τf (Z) defined in Definition8 and Definition 16.

Proposition 2.2.22. Let F = {f = 0} be a reduced hypersurface in Pn where f is a homogeneous polynomial in k[x0, ··· , xn] of degree d and let I be the ideal sheaf generated by the partial derivatives of f. Let z ∈ Z = V(I) then:

τ(Z, z) = τf (Z, z) and µ(Z, z) = µf (Z, z).

We postpone the proof of this result to Subsection 6.2.1.

Chapter 3

Resolution of the symmetric algebra

Given a smooth quasi-projective variety X of dimension n and an ideal sheaf I generated by n + 1 global sections of a line bundle L over X, we saw in Chapter2 n that the equations of the projectivization X of I in PX are determined by a locally free presentation of I. Namely, letting

M n+1 F OX I ⊗ L 0

n be a locally free presentation of I⊗L, the equations of X in PX are the entries of the row matrix y0 . . . yn M, where yi are the relative homogeneous coordinates n of PX . If F has rank n then X is the intersection of n divisor. Hence if X has dimension n the collection of entries in y0 . . . yn M is a regular sequence and n the resolution of the ideal I of X in PX is thus the Koszul complex associated X the sequence y0 . . . yn M, see Definition 2.2.5 for the definition of the Koszul complex. This fact is of great interest for us since it implies the vanishing of some higher direct image sheaves when pushing-forward the Koszul complex by the projection map, as we will see. However, in greater generality, i.e. when rank(F) > n, the Koszul complex does not resolve IX anymore. Our aim in this chapter is to prove that, provided that V(I) is zero-dimensional, the resolution of IX keeps a remarkable property which we call subregularity, see Definition 3.2.3. This property insures that the same type of vanishing of higher direct image sheaves holds as in the ”Koszul case”. Let us present more precisely our motivation with the following example. We refer to Section 1.2 and Subsection 1.3.3 for the associated definitions of Chow ring and to Subsection 3.1.1, Subsection 3.1.2 and Subsection 3.1.3 for the definitions of push-forward and Eagon-Northcott complex. This example is also a presentation of the problem in Chapter6.

2 Example 3.0.1. Let I be the ideal sheaf over OP generated by the 2 × 2-minors  2 x0 x0 2 of the matrix M = x1 x1 so that a locally free resolution of I reads: 2 x2 x2

51 52 CHAPTER 3. RESOLUTION OF THE SYMMETRIC ALGEBRA

3 0 O 2 (−1) ⊕ O 2 (−2) O 2 I(3) 0 P P P and a locally free resolution of the projectivization X of I reads

0, 0 OP(−3, −2) OP(−1, −1) ⊕ OP(−2, −1) OP OX

2 2 2 where P = P × P = P(OP [y0, . . . , yn]) and where we write to the left the shift in variables x0, . . . , xn and right the shift in the variables y0, . . . , yn. The naive 2 projective degree d2 of Φ is by definition the coefficient of h1 in the decomposition of X in the Chow ring of P and where h1 is the pull back of the hyperplane class of the first factor of P (see Definition 4.1.1 for the definition of naive projective degrees). In this case, the decomposition of X reads:

2 2 [X] = (h1 + h2)(2h1 + h2) = 2h1 + 3h1h2 + h2 since X is complete intersection and we have that d2 = 2. Equivalently, the naive projective degree d2 is the length of the scheme W deﬁned as the support of the cokernel of a general map O2 → O (0, 1) i.e. we have P X the following exact sequence:

O2 O (0, 1) O (0, 1) 0. (3.0.1) P X W

2 Let p1 : P → P be the ﬁrst projection. As we will explain in Subsection 3.1.1, we can push forward by p1 the sequence (3.0.1) to obtain the exact sequence

p O2 p O (0, 1) p O (0, 1). (3.0.2) 1∗ P 1∗ X 1∗ W

2 2 In this sequence, we have that p1∗O 'O 2 . Now let tentatively assume that P P p1∗OX(0, 1) 'I(3) and that the last map in (3.0.2) is surjective. As we will see this is not an obvious fact. Then we have the following exact sequence

2 O 2 I(3) p1∗OW (0, 1) 0 P

The length of p1∗OW (0, 1) is actually the length of a general cosection of the image sheaf E of the presentation matrix M. So provided that p1∗OX(0, 1) 'I(3) and that the last morphism in (3.0.2) is surjective, we can compute the 2nd naive topological degree on X. As we will explain, the two assumptions are veriﬁed since, under our hypothesis, the resolution of X over OP is subregular (the veriﬁcation of these two assumptions is precisely the problem of Chapter6).

3.1 Algebraico-geometric background

3.1.1 Cohomological results about direct image sheaves For this background about direct image sheaves, in particular about right derived functors, we refer to [Har77, III.1 Derived functors]. 3.1. ALGEBRAICO-GEOMETRIC BACKGROUND 53

Deﬁnition 3.1.1. Let p : X → Y be a continuous map of topological spaces. For any sheaf of abelian groups F on X, the direct image sheaf p∗F on Y is the sheaf −1 deﬁned by p∗F(V ) = F f (V ) for any open set V ⊂ Y .

Note that p∗ is a functor from the category Ab(X) of sheaves of abelian groups on X to the category Ab(Y ) of sheaves on Y . Actually, p∗ is left exact meaning that p∗ sends any exact sequence:

0 F 0 F F 00 0 (3.1.1) to the exact sequence

0 00 0 p∗F p∗F p∗F .

Moreover, Ab(X) has enough injectives, see [Har77, III.1]) so we can state the following definition: Definition 3.1.2. Let p : X → Y be a continuous map of topological spaces. i Then the higher direct image functors R p∗ : Ab(X) → Ab(Y ) is the right derived functors of p∗. For us the main use of the derived theory is as follows. The push forward of (3.1.1) gives by definition the long exact sequence:

0 00 1 0 1 0 p∗F p∗F p∗F R p∗F R p∗F 1 0 ... i 0 i i 0 ... R p∗F R p∗F R p∗F R p∗F where each of the sheaves involved can be computed thanks to the cohomology of F, F 0 and F 00. More precisely: Proposition 3.1.3. [Har77, III.8.5] Let X be a noetherian scheme, and let p : X → Y be a morphism from X to an aﬃne scheme Y = Spec(A). Then for any quasi-coherent sheaf F on X, we have

i i ∼ R p∗(F) ' H (X, F) .

The gain is that, in our context, we can compute the cohomology of sheaves thank to the two following theorems. Theorem 3.1.4. [Har77, III.2.7] Let X be a noetherian topological space of dimension n. Then for all i > n and all sheaves of abelian groups F on X, we have Hi(X, F) = 0.

n Theorem 3.1.5. [Har77, III.5.1] Let A be a noetherian ring, and let X = PA with n ≥ 1 be the projective space of dimension n over A. Then:

i (a)H X, OX (r) = 0 for 0 < i < n and all r ∈ Z,

n (b)H X, OX (−n − 1) ' A, 54 CHAPTER 3. RESOLUTION OF THE SYMMETRIC ALGEBRA

0 n n H X, OX (r) × H X, OX (−r − n − 1) → H (X, OX (−n − 1) ' A

is a perfect pairing of ﬁnitely generated free A-modules, for each r ∈ Z.

The last theorem is important because, most of the time, our framework is to have a product and two projections as in the following diagram:

X × Pn p1 p2 X Pn where X is a quasi-projective variety. In this setting, we consider push forwards ∗ ∗ p1∗(p1F ⊗ p2G) for F a locally free sheaf of OX -modules and G a locally free sheaf i ∗ ∗ n of OP -modules and we compute the right derived functors R p1∗(p1F ⊗ p2G) with the following result.

Proposition 3.1.6. Let X be a quasi projective variety and consider the product n X × P as in the previous diagram. Then, given a locally free sheaf F of OX - n modules and G a locally free sheaf of OP -modules, we have:

i ∗ ∗ i n R p1∗(p1F ⊗ p2G) = F ⊗ H (P , G)

Proof. First we use the projection formula [Har77, Ex.8.3] stating that

i ∗ ∗ i ∗ R p1∗(p1F ⊗ p2G) = F ⊗ R p1∗p2G

i ∗ i n We show now that R p1∗p2G = H (P , G) ⊗ OX . So we consider the following cartesian square

p n 2 X × P Pn

p1 f Spec(k) X g

By [Har77, III.9.3], there is a natural isomorphism

i n ∗ i i ∗ H (P , G) ⊗ OX = g R f∗G' R p1∗p2G

i i n since X is ﬂat over k. Now since R f∗G = H (P , G), we have

∗ i i n g R f∗G' H (P , G) ⊗ OX . 3.1. ALGEBRAICO-GEOMETRIC BACKGROUND 55

3.1.2 The Eagon-Northcott complex As an illustration of Subsection 3.1.1, we construct the resolution of the ideal of 3 the twisted cubic of P . Recall that the twisted cubic C = V I2(M) is the zero x x x locus of the 2 × 2-minors of the matrix M = 0 1 2 with entries in the x1 x2 x3 polynomial ring R = k[x0, . . . , xn]. As we explained in Subsection 2.1.1, we identify C with a complete intersection 3 1 3 1 2 scheme in the product = × , × being the projectivization O 3 . D P P P P P P P Hence a locally free resolution of OD reads:

3 3 0 OP(−3, −3) OP(−2, −2) OP(−1, −1) OP OD 0. (3.1.2)

∗ ∗ 3 1 The sheaves OP(−i, −j) are the sheaves p1OP (−i)⊗p2OP (−j) of Subsection 3.1.1 where p1 and p2 are the projections:

P3 × P1 p1 p2 P3 P1

Let us now decompose the push forward by p1 of the exact sequence (3.1.2). In order to do so, we denote as follows the kernel and cokernel in (3.1.2)

3 3 0 OP(−3, −3) OP(−2, −2) OP(−1, −1) OP OD 0

G2 G1

0 0 0

so that G1 and G2 are the kernel of (respectively) the ﬁrst and second homomor- phism of (3.1.2). Now, we apply p1∗ to the exact sequence:

0 G1 OP OD 0 to obtain 1 0 p1∗G1 p1∗OP p1∗OD R p1∗G1.

1 3 As we explained in Subsection 2.1.1, p1∗OD 'OC and p1∗OP 'OP so if R p1∗G1 = 0, a locally free resolution of OC is given by a locally free resolution of p1∗G1. We continue this computation by applying p1∗ to the exact sequence:

3 0 G2 OP(−1, −1) G1 0.

We obtain 56 CHAPTER 3. RESOLUTION OF THE SYMMETRIC ALGEBRA

3 1 0 p1∗G2 p1∗OP(−1, −1) p1∗G1 R p1∗G2

1 3 1 2 R p1∗OP(−1, −1) R p1∗G1 R p1∗G2.

3 1 3 Now we focus on p1∗OP(−1, −1) and R p1∗OP(−1, −1) . By Proposition 3.1.6, 3 0 1 0 1 3 1 1 p1∗OP(−1, −1) 'OP (−1) ⊗ H P , OP (−1) and H P , OP (−1) = 0 so

3 p1∗OP(−1, −1) = 0. In the same way,

1 3 1 1 3 1 R p1∗OP(−1, −1) 'OP (−1) ⊗ H P , OP (−1) and by Theorem 3.1.5 (c),

1 1 1 1 0 1 1 1 1 H P , OP (−1) = H P , OP (−(1 − 2) − 2) ' H P , OP (1 − 2) = 0.

1 2 Hence, we have that p1∗G2 = 0 and p1∗G1 ' R p1∗G2. Moreover, if R p1∗G2 = 0, 1 we have that R p1∗G1 = 0. Now we apply p1∗ to the exact sequence

3 0 OP(−3, −3) OP(−2, −2) G2 0 and we obtain

3 0 p1∗OP(−3, −3) p1∗OP(−2, −2) p1∗G2

1 1 3 1 R p1∗OP(−3, −3) R p1∗OP(−2, −2) R p1∗G2

2 2 3 2 R p1∗OP(−3, −3) R p1∗OP(−2, −2) R p1∗G2

3 R p1∗OP(−3, −3).

In this sequence, p1∗G2 = 0 as we have already explained. We compute the sheaf 1 R p1∗OP(−3, −3) as follows. By Proposition 3.1.6,

1 1 1 3 1 R p1∗OP(−3, −3) 'OP (−3) ⊗ H P , OP (−3)

1 1 0 1 1 1 and by Theorem 3.1.5 (c),H P , OP (−3) ' H P , OP (1) which has dimension 2. So 1 2 3 R p1∗OP(−3, −3) 'OP (−3) and, in the same way,

1 3 3 3 R p1∗OP(−2, −2) 'OP (−2) . 3.1. ALGEBRAICO-GEOMETRIC BACKGROUND 57

Using Theorem 3.1.4, we compute also that the higher direct image sheaves 1 vanish. Hence, a locally free resolution of R p1∗G2 reads

2 3 1 3 3 0 OP (−3) OP (−2) R p∗G2 0 so that a locally free resolution of OC is

2 3 3 3 3 0 OP (−3) OP (−2) OP OC 0.

Actually, this situation consisting of pushing forward a Koszul complex in order to describe a resolution of the coordinate ring of a determinantal variety is much more general. The ﬁnal complex is called the Eagon-Northcott complex. We give its deﬁnition following [Eis05, A2H]. Let R be a ring and let F = Rf and G = Rg be two free R-modules. The Eagon-Northcott complex of a map α : F → G (or a matrix A representing α) is a complex

df−g+1 df−g ∨ f ∨ f−1 EN(α): 0 Symf−g G ⊗ ∧ F Symf−g−1 G ⊗ ∧ F

g d3 d2 ∧ α ... ∨ g+2 ∨ g+1 g g Sym2 G ⊗ ∧ F G ⊗ ∧ F ∧ F ∧ G.

Here Symk G is the k-th symmetric power (or divided power in the case that k has positive characteristic, see[Eis95, A.2.4]) of G. The homomorphisms dj are deﬁned as follows. First, we deﬁne a diagonal map

∨ ∨ ∨ ∆ : Symk G → G ⊗ Symk−1 G as the dual of the multiplication map G ⊗ Symk−1 G → Symk G in the symmetric algebra of G. Next we deﬁne an analogous map

∆ : ∧kF → F ⊗ ∧k−1F as the dual of the multiplication in the exterior algebra of F ∨. On decomposable elements, this diagonal has the simple form

X i−1 ˆ f1 ∧ ... ∧ fk 7→ (−1) fi ⊗ f1 ∧ ... ∧ fi ∧ ... ∧ fk. i

∨ P 0 00 ∨ ∨ For u ∈ Symj−1 G we write ∆(u) = i ui⊗ui ∈ G ⊗Symj−2 G and similarly for g+j−1 P 0 00 g+j−2 ∨ 0 ∨ v ∈ ∧ F we write ∆(v) = t vt ⊗ vt ∈ F ⊗ ∧ F . Note that α (uj) ∈ F ∨ 0 0 so [α (ui)](vt) ∈ R. We set

∨ g+j−1 ∨ g+j−2 dj: Symj−1 G ⊗ ∧ F Symj−2 G ⊗ ∧ F P ∨ 0 0 00 00 u ⊗ v [α (us)](vt) · us ⊗ vt s,t 58 CHAPTER 3. RESOLUTION OF THE SYMMETRIC ALGEBRA

We refer to [Eis05, A2H] for the fact that EN(α) is a complex. The following theorem provides a condition under which EN(α) is resolution of coker(∧gα). Re- call that the depth of an ideal I over a ring R is the length of a maximal regular sequence o R in I.

Theorem 3.1.7. [Eis05, A.2.60] Let α : F → G with rank(F ) = f ≥ rank(G) = g be a map of free R-modules of ﬁnite rank. The Eagon-Northcott complex EN(α) is exact (and thus furnishes a free resolution of R/Ig(α)) if and only if depth Ig(α) is equal to f − g + 1, the greatest possible value.

More details about the construction of the Eagon-Northcott complex as a push- forward of a Koszul complex can be found in [Eis95, A2] or [Wey03, 6]. The idea is exactly the one we applied in the case of the twisted cubic of P3 in the beginning of the subsection. Namely, denoting by C the scheme V Ig(α) , we have that C is isomorphic to a complete intersection D in P(G). The push forward of the resolution of OD, that is a Koszul complex, provides a resolution of OC. This is the Eagon-Northcott complex resolving OC.

3.1.3 Gorenstein rings and linkage To ﬁnish this background section, we introduce the notions of Gorenstein rings and linkage. To this end, we focus one more time on the twisted curve C in P3. 2 Recall that the ideal sheaf IC of C is generated by the sections x0x2 − x , x0x3 − 1 2 x0 x1 x2 x1x2, x1x3 − x2 which are the 2 × 2-minors of the matrix M = . x1 x2 x3 2 Now consider the sub-ideal sheaf IC∪L generated by x0x2 −x1, x0x3 −x1x2. It is a computation to show that V(IC∪L) is the union of the twisted cubic C and a line 3 L = V(x0, x1) in P . Actually, C and L meet in one point p of multiplicity 2 (i.e. length(OC∪L,p) = 2). In this case, since the union C ∪ L is a complete intersection curve, we say that C and L are linked. What is remarkable in this example is that the quotient ideal (IC∪L : IC) is equal to IL and (IC∪L : IL) = IC. This property of linkage of IC∪L is a property shared by any complete intersection scheme. It is the property of being Gorenstein. Let us now give more precise deﬁnitions and results concerning this domain following [Eis95, 21].

Deﬁnition 3.1.8. Let A be a local noetherian ring of Krull dimension n, m its maximal ideal and k its residue ﬁeld. A is called Gorenstein if

i n ExtA(k,A) = 0 for all i 6= n and Ext (k,A) ' k.

A scheme is called Gorenstein if all its local rings are Gorenstein.

Deﬁnition 3.1.9. Let A be a local noetherian ring of Krull dimension n, m its maximal ideal and k its residue ﬁeld. An A-module M such that the module n ExtA(A/m,M) vanishes if n 6= height(m) and is 1-dimensional if n = height(m) is called the canonical module of A. In this case, we denote ωA the canonical module M.

As a ﬁrst consequence, we have. 3.1. ALGEBRAICO-GEOMETRIC BACKGROUND 59

Proposition 3.1.10. In the situation where A has a canonical module, A is Goren- stein if and only if ωA = A. Given a Gorenstein scheme X, the invertible sheaf associated to the local canonical modules is called the dualizing sheaf of X and denoted by ωX . Theorem 3.1.11. [Eis95, 21.19] Any complete intersection scheme is Gorenstein. As we will see, determinantal schemes are also Gorenstein under certain conditions. For us, the result we will use the most concerning Gorenstein schemes is Theorem 3.1.14 below. Recall that the codimension of an ideal I over a ring A is the Krull dimension of the localization AI [Eis95, 9]. We deﬁne also Cohen-Macaulay ideal. Deﬁnition 3.1.12. A ring R is Cohen-Macaulay if depth(I) = codim(I) for every ideal I of R. Proposition 3.1.13. [BH93] Gorenstein rings are Cohen-Macaulay. Theorem 3.1.14. [Eis95, 21.23] Let A be a Gorenstein local ring, and let I be an ideal of codimension 0. Set B = A/I and J = (0 :A I). We have J ' HomA(B,A). (a) The ideal J has codimension 0 and no embedded components. If I has no embedded components, then I = (0 :A J) so I and J are linked. (b) If B = A/I is a Cohen-Macaulay ring, then C = A/J is a Cohen-Macaulay ring.

(c) If B = A/I is a Cohen-Macaulay ring, then J = (0 :A I) = HomA(B,A) is a canonical module for B; in particular, B is Gorenstein if and only if J is a principal ideal of A. To ﬁnish this section about Gorenstein properties, let us explain in more details some facts we will use in the following.

Proposition 3.1.15. [Eis95, Theorem 21.15] Let P be a smooth variety, ωP be its dualizing sheaf and let K be a Gorenstein subscheme of P of codimension n. Then n the dualizing sheaf ωK of K veriﬁes ωK 'Ext (OK, ωP). Remark 3.1.16. In the settings of Proposition 3.1.15, let

Q Q ... 0 Qn n−1 n−2 Q0 OK 0

En−1 En−1 E1

be a locally free resolution of OK, it has length n, since, as a Gorenstein scheme, K is Cohen-Macaulay. By this, we set that the sheaves Qi are the locally free sheaves of the resolution and the sheaves Ei are the kernels and cokernels of the corresponding morphisms. Then, a locally free resolution of ωK reads:

∨ ∨ ∨ ... ∨ 0 Q0 Q1 Q2 Qn ωK 0. 60 CHAPTER 3. RESOLUTION OF THE SYMMETRIC ALGEBRA

Indeed, applying the functor Hom(·, ωP) successively on each exact sequence

0 Ei Qi−1 Ei−1 0

1 n we can compute by cohomology chasing that Ext (En−1, ωP) 'Ext (OK, ωP) ' ωK. j This is because the derived sheaves Ext (Qi, ωP) vanishes for all i ∈ {0, . . . , n} and all j > 0 since the sheaves Qi are locally free.

3.2 Subregularity of the symmetric algebra

3.2.1 Generalities

Assuming that the subscheme Z = V(IZ ) of X deﬁned by IZ is zero-dimensional in a quasi-projective variety X, we provide here a resolution of the ideal of X = P(IZ ) in terms of the pulled-back resolution of the dualizing module of Z, up to some shift in degree, and of the Eagon-Northcott complex associated with another still larger algebra, which we call the Koszul hull. Our precise settings are as follows unless otherwise speciﬁed.

Notation 3.2.1. Put n = dim(X) and let IZ be an ideal sheaf generated by n + 1 global sections φ0, . . . , φn of a vector bundle L over X. We assume that the subscheme Z = V(IZ ) of X is 0-dimensional. Letting V be the (n+1)-dimensional subspace generated by φ0, . . . , φn, we consider a surjection

∨ V ⊗L → IZ → 0 (3.2.1)

n which provides a closed embedding X ,→ P ' PX of X = P(IZ ). Here, we denote by P the projective bundle n+1 Proj Sym(OX (−η) ) where η stands for c1(L). We also consider its bundle map p : P → X and we let y0, . . . , yn be its relative homogeneous coordinates. ∗ Depending on the context η can stand also for p c1(L) and we let ξ be the ﬁrst

Chern class of OP(1). Now, put

i+1 Qi,j = ( ∧ V) ⊗ OP(−(j + 1)ξ − (i − j)η) for 1 ≤ i ≤ n and 0 ≤ j ≤ i − 1

i−1 and Qi = ⊕ Qi,j. The sheaves Qi are the terms of the Eagon-Northcott complex j=0 associated with the map

ψ :V ⊗O → O (η) ⊕ O n (ξ). P P PX deﬁned by the matrix φ . . . φ 0 n . y0 . . . yn The complex takes the form: 3.2. SUBREGULARITY OF THE SYMMETRIC ALGEBRA 61

0 Q ... Q O n . (Q ) n 1 PX • Since dim(Z) = 0, let:

(P ) 0 Pn ... P1 P0 OZ 0 •

∨ be a locally free resolution of OZ , so here P0 = OX and P1 = V ⊗L . We emphasize that the length of this resolution is n = dim(X) since the scheme Z is locally Cohen-Macaulay. Set 0 ∗ ∨ Pi = p Pn+1−i ⊗ OP(−nη − ξ) for 1 ≤ i ≤ n + 1 and let IX be the ideal of X into P. Our result is the following: Theorem 3.2.2. Under the assumption that dim(Z) = 0, X is Cohen-Macaulay of dimension n and there is a locally free resolution of IX of the following form:

Qn Q1 0 0 Pn+1 ⊕ ... ⊕ IX 0. (R1) 0 0 Pn P1

Denoting by yi the homogeneous relative coordinates of the projective bundle P, we make the following deﬁnition.

Definition 3.2.3. A complex (R•) over P is subregular if for all i the differential Ri → Ri−1 is linear or constant in the y variables. In the proof of Theorem 3.2.2 page 70, we will see that the differentials of R1 are actually linear or constant in the y variables so, anticipating, we can state that:

Corollary 3.2.4. The ideal IX admits a subregular locally free resolution over P. In the last subsection, we focus on a graded version of this result. The motivation to study speciﬁcally this case is that it is the framework when working with a rational map from Pn to Pn. It is moreover the case in which we have a more reﬁned minimal free resolution of the symmetric algebra. For instance

2 2 Example 3.2.5. Let IZ = (x1 − x1x3, x2 − x2x3, x1x2, x0x3) be the ideal over 3 R = k[x0, . . . , x3] (remark that that Z = V(IZ ) is indeed 0-dimensional in P ). By a computation via Macaulay2, we have that a minimal free resolution of IZ reads

R(−3)2 2 4 0 R(−5) ⊕ R(−2) IZ 0. R(−4)3

Now, take the last two modules of this complex and tensor them by S(2, −1) where S = R[y0, . . . , y3]. They give the beginning of a minimal free resolution of the ideal IX of the symmetric algebra of IZ over S (also computed with Macaulay2) 62 CHAPTER 3. RESOLUTION OF THE SYMMETRIC ALGEBRA

S(−4, −2) S(−2, −2)4 S(−1, −1)2

0 ⊕ ⊕ ⊕ IX 0. S(−2, −3) S(−3, −1)2 S(−2, −1)3

More generally take R = k[x0, . . . , xn] and IZ = (φ0, . . . , φn) an ideal over R generated by n + 1 homogeneous polynomials of degree η (this integer η should be n understood as the degree of c1(L) in the sheaﬁﬁed case assuming X = P ). The ideal of the symmetric algebra of IZ , denoted by IX, is a bigraded homogeneous 0 ideal of S = R[y0, . . . , yn]. This time we consider the two complexes (P•) and (Q•) 0 obtained by taking the graded modules of global sections of (P•) and (Q•). These are S-graded subregular complexes. Our result in this setting is the following.

Theorem 3.2.6. Assume IZ is a graded homogeneous Cohen-Macaulay ideal of dimension 1, then X is Cohen-Macaulay and a minimal bigraded S-free resolution of IX reads:

00 00 00 Qn−1 Qn−2 Q2 00 00 0 Qn ⊕ ⊕ ... ⊕ P1 IX 0 (R2) 00 00 00 Pn−1 Pn−2 P2 where

n n+1 00 ( i+1 ) 00 Qi = ⊕ Qi,j,Qi,j = S − (i − j)η, −j − 1) ,Pi = Pi+1 ⊗ S(η, −1). j=1

Moreover Theorem 3.2.2 and Theorem 3.2.6 are sharp in the following sense. If dim(Z) > 0, then the resolution of X might not be subregular as shown in the following example explained to us by Aldo Conca.

3 3 Example 3.2.7. In P , consider the zero locus Z of the ideal IZ = (−x2x3 + 4 4 4 3 4 2 2 4 x3, −x2 − x3, −x1x3 − x3, x2x3 + x3). The ideal IZ has Krull dimension 2 over R = k[x0, . . . , x3], so dim(Z) = 1, and a minimal graded free resolution of IX reads:

S(−4, −1) ⊕ S(−1, −1) S(−5, −2) S(−3, −2)3 ⊕ 2 0 S(−5, −3) ⊕ ⊕ S(−2, −1) IX 0 S(−4, −3)3 S(−4, −2) ⊕ ⊕ S(−3, −1) S(−3, −3) where we wrote the shift in the y variables in the right position. Hence the resolution of IX is not subregular. 3.2. SUBREGULARITY OF THE SYMMETRIC ALGEBRA 63

3.2.2 Local resolution of the symmetric algebra Preliminaries

Recall Proposition 2.1.3 that letting a locally free presentation of IZ

M P2 → P1 → IZ → 0, (3.2.2)

X is the zero scheme of the corresponding section of the composition map s ∈ 0 ∗ ∨ H P, p P2 ⊗ OP(ξ) . In other words, the ideal sheaf IX of X in P is locally ∗ generated by the entries of the row matrix yp M where y stands for (y0 . . . yn). We denote by Mx the matrix obtained from M by specializing at the point x ∈ X. We emphasize the following remark. Since dim(X) = n and codim(Z,X) = n, the local ideal sheaf IZ,z of a point z ∈ Z is generated by at least n independent sections of L lying in V. The crucial point is to take care of the case where z ∈ Z is a point at which Z is not a complete intersection, i.e all the sections φ0, . . . , φn are required to generate IZ,z. The following result is restatement of Proposition 2.2.13 but we re-explain its proof in the current settings.

Lemma 3.2.8. Let x ∈ X be a closed point. The scheme-theoretic ﬁbre Xx is:

(i) a point if x∈ / Z,

n−1 (ii) isomorphic to Px if x ∈ Z and Z is a local complete intersection at x,

n (iii) isomorphic to Px if x ∈ Z and Z is not a local complete intersection at x.

Proof. Since the formation of the symmetric algebra commutes with base change (see Proposition 2.1.1), the ﬁbre Xx is obtained by localizing X at x and taking P(IZ ⊗ kx), where kx is the residue ﬁeld of OX at x.

(i) If x∈ / Z, locally at x the ideal IZ is just OX , so p is an isomorphism of Xx to x.

(ii),(iii) If x ∈ Z, since Z has codimension n in X, a subspace of n independent local sections of L from V is needed at least to generate IZ locally around x. Actually such subspace exists if and only if Z is a local complete intersection (LCI) at x. In other words, IZ ⊗ kx is a kx-vector space which can be generated by an n-dimensional subspace of V if and only if Z is LCI at x, so n n+1 that IZ ⊗ kx is isomorphic to kx or to kx depending on whether Z is LCI n−1 n at x or not. Therefore P(IZ ⊗ kx) is isomorphic to Px or Px depending on whether Z is LCI at x or not.

Remark 3.2.9. Recall that in our setting of a zero-dimensional scheme Z, the set n of points z ∈ Z such that Xz ' Pz is equal set theoretically to V Fittn−1(IZ ) where Fittn−1(IZ ) is the ideal generated by the entries of M (Proposition 2.2.19). 64 CHAPTER 3. RESOLUTION OF THE SYMMETRIC ALGEBRA

Φ Now, we take the Koszul complex with respect to the map V ⊗OX (−η) = P1 −→ OX and we write ki(Φ) for the i-th diﬀerential of the Koszul complex. We have the following diagram:

2 k1(Φ) Φ ∧ P1 P1 IZ 0 (K3) F1 0 0 in which the first row is not exact since Z is not empty and where we put F1 = Im k1(Φ) . By definition of the presentation and the Koszul complex, we have F1 ⊂ E and E/F1 = H1(IZ ) where E is the kernel of the morphism P1 → IZ and H1(IZ ) stands for the first Koszul homology sheaf of the set (φ0 . . . φn) of generators of IZ (see Definition 2.2.5 for the definition of the Koszul homology).

The Koszul hull

We introduce now another subscheme of P which we call the Koszul hull of X. This n subscheme contains X and actually diﬀers from X by a copy of PZ , as we will see.

Deﬁnition 3.2.10. Set notation as in (K3) and let IK be the ideal sheaf generated ∗ by the entries in the row matrix yp k1(Φ). We call the Koszul hull, denoted by K, the subscheme in P deﬁned by K = V(IK).

Now, we explain the strategy of the proof of Theorem 3.2.2. Via the inclusion

F1 ⊂ E, we see that IK ⊂ IX, that is X ⊂ K. Hence we have the following short exact sequence:

0 IK IX IX/IK 0.

So in order to get the subregularity of the resolution of IX, we ﬁrst show the subregularity of resolutions of IK and of IX/IK and from there, we show how we get the resolution of IX by patching together these resolutions. We start by analysing the Koszul hull more closely.

Proposition 3.2.11. We have the following properties.

(i) The scheme K is determinantal. More precisely, IK is the ideal of the 2 × 2 minors of the map V ⊗OP → OP(η) ⊕ OP(ξ) deﬁned by the matrix:

φ . . . φ ψ = 0 n . y0 . . . yn

Under the assumption that dimX (Z) = 0:

(ii) codim(K, P) = n. 3.2. SUBREGULARITY OF THE SYMMETRIC ALGEBRA 65

(iii) A locally free resolution of IK is the sheaﬁﬁcation of the Eagon-Northcott complex. Namely, there is a long exact sequence:

0 Qn ... Q2 Q1 IK 0 (Q•)

i−1 where Qi = ⊕ Qi,j and j=0

i+1 Qi,j = ( ∧ V) ⊗ OP(−(j + 1)ξ − (i − j)η) for 1 ≤ i ≤ n and 0 ≤ j ≤ i − 1.

(iv) The scheme K is Gorenstein and its dualizing sheaf ωK veriﬁes: ∗ ωK ' p ωX ⊗ OP(nη − nξ).

Proof. (i) The morphism k1(Φ) takes the form,   φ1 φ2 ... −φ0 0 ...    0 −φ0 ... k1(Φ) =    .   . 0 ...  . .  . . ...

∗ and IK is generated by the entries in the row matrix yp k1(Φ). Those entries are the same as the 2 × 2 minors of the matrix ψ.

(ii) We argue set-theoretically by looking at the fibres of the map K → X obtained as restriction of p to K. First, note that if z 6∈ Z, then it is clear by the definition of K that Kz is a single point. On the other hand, if z ∈ Z n then φi(z) = 0 for all i ∈ {0, . . . , n} so by definition of K we have Kz = Pz . n Therefore the reduced structure of K is the union of X and of ∪z∈Z Pz . This proves that K has dimension n. (iii) Since K is determinantal of the expected codimension, it is Cohen-Macau- lay [BV88, Cor. 2.8]. Hence depth(IK) = codim(K, P) = n. Therefore the Eagon-Northcott complex provides a global resolution of the ideal IK [BV88, 2 2 Th. 2.16]. The first map ∧ V ⊗OP → ∧ OP(η) ⊕ OP(ξ) of the Eagon- 2 Northcott complex is the matrix ∧ ψ. Hence the complex (Q•) provides a

resolution of IK. (iv) Refering to Proposition 3.1.15, we know that if K is Gorenstein, then it- n s dualizing sheaf ωK should be isomorphic to Ext (OK, ωP). So set ωK = n Ext (OK, ωP) and let show that this veriﬁes the properties of a dualizing sheaf of a Gorenstein scheme. By the previous item (iii), a resolution of ωK is given by:

∨ ∨ M1 ∨ 0 Q1 ⊗ ωP ... Qn−1 ⊗ ωP Qn ⊗ ωP ωK 0 (3.2.3)

see Remark 3.1.16 for a better explanation of this fact. Hence, the ﬁrst i condition on the vanishing of the sheaves Ext (OK, ωK) for i < n is veriﬁed 66 CHAPTER 3. RESOLUTION OF THE SYMMETRIC ALGEBRA

i locally by localizing (3.2.3) at any p ∈ K and noting that Ext (OK,p, ωK, p) vanishes.

We show now that ωK is locally free of rank 1. Locally, we can write explicitly the matrix M1 which is the transpose of the last matrix in the Eagon- Northcott complex. So M1 has size n × (n − 1)(n + 1) and locally takes the form:   φ0 φn 0 0    y0 yn φ φ   0 n 0 0    M1 = 0 0 y0 yn φ0 φn 0 0      0 0 y0 yn

Consider an open cover of X by a family of open subsets {Ut | t ∈ J} such that Ut ∩ Z = {zt}. If z ∈ U ⊂ Ut\{zt} for all t, then the restriction of IZ to U is equal to OU so that KU = XU = U is obviously Gorenstein, because U is smooth.

Or else, if z = zt for some t, then φs(z) = 0 for all s ∈ {0, . . . , n}. In this case, n since every point (y0 : ... : yn) ∈ Pz has at least one non zero coordinate, the matrix (M1)z has corank 1. This shows that for any point of X, the stalk of ωK has rank 1 at that point, so ωK is locally free of rank one. Hence KUj is Gorenstein. This proves that K is Gorenstein. Now, we show the isomorphism

∗ ωK ' p ωX ⊗ OP(nη − nξ).

To do this, we ﬁrst give an explicit formula for ωK by describing the scheme K as a complete intersection into a larger projective bundle (see [Ein93] for more details about this construction). Let B be the projective bundle P O (η) ⊕ P OP(ξ) and put ζ for the relative hyperplane class of the bundle map q : B → P. A divisor D in |OB(ζ)| corresponds to a morphism ψD : OP → OP(η) ⊕ OP(ξ). Since the matrix ψ whose 2 × 2 minors deﬁne K has constant rank 1 over K, the map q restricts to an isomorphism from the complete n intersection ∩i=0Di to K, where Di corresponds to ψDi = (φi, yi). Therefore, by adjunction we have:

∗ q ωK ' ωB (n + 1)ζ . (3.2.4)

Next, we show that:

OK(ζ) 'OK(η). (3.2.5)

Indeed, given a divisor D ∈ |OB(ζ)|, the intersection D ∩ K is deﬁned in P by the vanishing of the 2 × 2 minors of the matrix:

φ . . . φ φ 0 n D , y0 . . . yn yD 3.2. SUBREGULARITY OF THE SYMMETRIC ALGEBRA 67

where ψD = (φD, yD) corresponds to D. Since yD lies in hy0, . . . , yni, this matrix is equivalent up to row and column operations to:

φ . . . φ φ0 0 n D , y0 . . . yn 0

0 0 for some φD ∈ H (X, L). 0 0 This means that the ideal of D ∩ K in K is generated by (y0φD, . . . , ynφD). Since all the yi do not vanish simultaneously, this implies that OK(ξ) is 0 generated by the restriction to K of φD. Hence OK(ζ) 'OK(η) and we compute: ∗ ωP ' p ωX ⊗ OP − (n + 1)ξ and therefore: ∗ ωB ' q ωP ⊗ OB(−2ζ + η + ξ).

∗ Hence by (3.2.4) and (3.2.5), we get that ωK ' p ωX ⊗ OP(nη − nξ).

From the description of the morphisms di of the Eagon-Northcott complex given just before Theorem 3.1.7, we have that these morphisms have constant or linear entries in the yi variables. Hence

Corollary 3.2.12. The resolution (Q•) is subregular.

Description of the quotient IX/IK

We show now the subregularity of a locally free resolution of the quotient IX/IK. Let us outline that since Z is a zero dimensional scheme of ﬁnite type it is aﬃne hence projective. As such it has a dualizing sheaf ωZ , see [Har77, Proposition 7.5].

Proposition 3.2.13. We have the following isomorphism:

∗ ∨ IX/IK ' p (ωZ ⊗ ωX ) ⊗ OP(−nη − ξ).

The proof of this proposition is the object of Lemma 3.2.14. Its proof and the proof of Proposition 3.2.13 rely mostly on Theorem 3.1.14.

∗ Lemma 3.2.14. The quotient ideal sheaf (IK : IX) is isomorphic to p IZ .

Proof. As in the proof of Proposition 3.2.11, we denote by k1(y) the ﬁrst diﬀerential in the Koszul complex associated to the map (y0 . . . yn). We denote also by IX,K ∗ n the ideal of X in K and W stands for the scheme p Z. Of course we have W ' PZ . The inclusion IK ⊂ IW explains the right horizontal exact sequence in the following commutative diagram: 68 CHAPTER 3. RESOLUTION OF THE SYMMETRIC ALGEBRA

∗ yp k1(Φ) 2 ∧ V ⊗OP(−ξ − η) IK 0

k1(y) p∗M p∗Φ ∗ 0 p E V ⊗OP(−η) IW 0 y β

0 C OP(ξ − η) IW/IK 0

0 0 0

The commutativity in the right above square comes from the following fact. Writing down the matrix k1(y) as follows:   y1 y2 ... −y0 0 ...    0 −y0 ... k1(y) =    .   . 0 ...  . .  . . ...

∗ ∗ and similarly for k1(Φ), it is direct computation to show that yp k1(Φ) = p Φk1(y). ∗ Hence, the image of the map β = yp M is exactly the ideal IX(ξ) and we have that:

Ann(IW/IK) 'IX. Now we use the assumption that Z is zero-dimensional. Since the statement is local and the formation of the symmetric algebra commutes with base change (see

Proposition 2.1.1), we can assume that OP and OK are Gorenstein local rings. We apply Theorem 3.1.14 to the Gorenstein scheme K and to the ideal sheaf IX,K. We denote by IW,K the ideal of W in K. Since W has codimension 0 in K and has no embedded components, the ideals IW,K and IX,K are linked in OK. This shows that IW,K = Ann(IX,K). Now, since we have already IK ⊂ IW, the equality occurs as ideal sheaves of OP itself. Moreover we have the isomorphism Ann(IX,K) ' (IK : IX). Hence:

∗ IW = p IZ ' (IK : IX).

Proof of Proposition 3.2.13. As above, we can assume that OP and OK are Goren- stein local rings and we apply Theorem 3.1.14 to OK. We denote again by IX,K the ∗ ideal of X in K and by IW,K the ideal of W in K (recall that W = p Z). Since IX,K has codimension 0 in OK, we have that (IK : IX) and IX,K are linked. But following the notation in Lemma 3.2.14,(IK : IX) 'IW,K. 3.2. SUBREGULARITY OF THE SYMMETRIC ALGEBRA 69

Moreover, applying Theorem 3.1.14 in a sheaﬁﬁed case, W is Cohen-Macaulay as a pull back of Z so X is also Cohen-Macaulay and we have: I ' ω ⊗ ω∨ W,K X K where ωX is the canonical sheaf of X. We also have that:

ω ⊗ ω∨ 'I 'I /I . W K X,K X K n ∗ Now, since W ' PZ , we have ωW ' p ωZ ⊗ OP(−(n + 1)ξ). Therefore, by Proposition 3.2.11:

I /I ' ω ⊗ ω∨ ' p∗(ω ⊗ ω∨ ) ⊗ O (−nη − ξ). X K W K Z X P

Denoting H1(IZ ) the ﬁrst Koszul homology associated to

Φ:V ⊗OX (−η) → OX , as in (K3), we emphasize the following point in order to elucidate the nature of the sheaf IX/IK.

Proposition 3.2.15. The sheaf IX/IK is isomorphic to the pull-back of the ﬁrst homology H1(IZ ) of Φ up to a shift. More precisely, we have

∗ IX/IK ' p H1(IZ ) ⊗ OP(η − ξ).

Proof. To shorten the notation, we set H1 for H1(IZ ). We are going to show that

∨ H1 ' ωZ ⊗ ωX −(n + 1)η . (3.2.6)

n First, ωZ 'Ext (OZ , ωX ). Hence, we will prove (3.2.6) by showing that

n ∨ OZ 'Ext (H1, ωX ) ⊗ ωX −(n + 1)η .

To this end, let:

n+1 kn(Φ) 2 k1(Φ) Φ 0 ∧ P1 ... ∧P1 P1 IZ 0 (K4)

F2 F1 ⊂ E

i i be the Koszul complex associated with Φ = (φ0 . . . φn), where ∧P1 = (∧ V) ⊗ OX (−iη). Since codim(Z,X) = depth(IZ ) = n the Koszul homology is concen- trated in degree 1 and by deﬁnition H1 = E/F1. Applying the functor Hom(−, ωX ) to (K4), we obtain:

0 Hom(F1, ωX ) ... V ⊗ωX (nη)

1 ωX (n + 1)η Ext (Fn−1, ωX ) 0 70 CHAPTER 3. RESOLUTION OF THE SYMMETRIC ALGEBRA

1 n−1 and it is a computation to show that Ext (Fn−1, ωX ) 'Ext (F1, ωX ). n−1 n The last point is that Ext (F1, ωX ) 'Ext (H1, ωX ). Indeed, by the long exact sequence associated to the short exact sequence:

0 F1 E H1 0 we have the following exact sequence:

n−1 n−1 n n Ext (E, ωX ) Ext (F1, ωX ) Ext (H1, ωX ) Ext (E, ωX )

n−1 n and Ext (E, ωX ) = Ext (E, ωX ) = 0 since Z is locally Cohen-Macaulay. Moreover, the last map kn(Φ) of the Koszul complex is the transpose of the ﬁrst map Φ up to signs. Thus the maps in the sequence:

n V ⊗ωX (nη) −→ ωX (n + 1)η → Ext (H1, ωX ) → 0 are the same as the maps in the exact sequence:

Φ P1 −→OX → OZ → 0.

n Taking care of the twisting, this means that OZ ⊗ ωX (n + 1)η 'Ext (H1, ωX ). ∨ This implies H1 ' ωZ ⊗ ωX −(n + 1)η . 0 Remark 3.2.16. To enlighten the construction of the sheaves Pi for i ∈ {1, . . . , n+ 1} in the following proof of Theorem 3.2.2, recall that the complex:

0 Pn ... P1 P0 OZ 0 (P•) is a locally free resolution of OZ . Hence, applying the functor Hom(−, ωX ) to Equation (P•), a locally free resolution of ωZ reads:

∨ ∨ 0 P0 ⊗ ωX ... Pn ⊗ ωX ωZ 0

(see Remark 3.1.16 for a better explanation of this fact) from which we can read a ∨ locally free resolution of ωZ ⊗ ωX . Proof of Theorem 3.2.2. As we saw in Lemma 3.2.8 and in the proof of Proposi- tion 3.2.13, X is Cohen-Macaulay of dimension n. Moreover, by Proposition 3.2.11 and Proposition 3.2.13, we have the following commutative diagram:

... 0 Qn Q2 Q1 IK 0

0 0 ... 0 0 0 Pn+1 Pn P2 P1 IX/IK 0.

0 3.2. SUBREGULARITY OF THE SYMMETRIC ALGEBRA 71 where i−1 i+1 Qi = ⊕ ( ∧ V) ⊗ O (−(j + 1)ξ − (i − j)η) j=0 P and 0 ∗ ∨ Pi = p Pn+1−i ⊗ OP(−nη − ξ) for 1 ≤ i ≤ n + 1. Let us explain now how these resolutions patch together to give the desired resolution of IX. So denote Ki the kernels (and cokernels) of the complex (Q•) and Ti as in the following diagram :

K2 K1 0

... 0 Qn Q2 Q1 IK 0 dn d2 d1

IX δ δ δ δ 0 n+1 0 n ... 0 2 0 1 0 Pn+1 Pn P2 P1 IX/IK 0

Tn T2 T1 0 where δi and di stands for the morphisms of the associated resolutions. In order to ˜ 0 construct the desired resolution, we first prove that δ1 lifts to a map δ1 : P1 → IX . 1 0 1 0∨ To do this, it suffices to prove that Ext P1, IK = 0, that is H P, IK ⊗ P1 = 0. ˜ ˜ Observe that, once we show that δ1 exists, then the map (d1, δ1): Q1 ⊕ P1 → IX is surjective. Let us now check the required vanishing. In view of (Q•), it suffices i 0∨ to show that H (P, Qi ⊗ P1 ) = 0 for all i ∈ {1, . . . , n}. Proposition 3.1.6 implies i n n these vanishings since H P , OP (−j) = 0 for all j = 0, . . . , i − 1. In the case i = n, we use that n n 0 n n n H P , OP (−j) ' H P , OP (j − n − 1) and the fact that j − n − 1 ≤ −2. 0 Now consider the kernel S1 of the constructed morphism Q1 ⊕ P1 → IX. We have an exact sequence

0 K1 S1 T1 0.

0 0 Next, we construct a surjection Q2 ⊕P2 → S1 again by lifting the map δ2 : P2 → T1 ˜ 0 1 0 to δ2 : P2 → S1. This is achieved if we show that Ext (P2, K1) = 0. Again, we use a piece of (Q•) to show this vanishing. Indeed, we write the resolution

0 Qn ... Q2 K1 0. Applying Proposition 3.1.6 to this complex yields again the required vanishing. ˜ 0 Therefore, we have now a surjection (d2, δ2): Q2 ⊕ P2 → S1. Composing with the 0 injection of S1 into Q1 ⊕ P1, by construction we get   d1 a1 Q   Q 2 0 δ1 1 ⊕ −→ ⊕ 0 0 P2 P1 72 CHAPTER 3. RESOLUTION OF THE SYMMETRIC ALGEBRA

˜ where a1 is the composition of δ2 with the natural map S1 → Q1. Note that the map a1 can have only constant or linear entries with respect to the yi variables because of degree reasons. Continuing this way we construct a resolution

Qn Q1 0 0 Pn+1 ⊕ ... ⊕ IX 0. (3.2.7) 0 0 Pn P1 of IX. The morphisms in the above resolution take the form   di ai Q   Q i+1 0 δi i ⊕ −→ ⊕ 0 0 Pi+1 Pi where i ∈ {1, . . . , n}. These morphisms have linear or constant entries as this holds for δi (as di of course does not depend on the yj), for ai (by degree reasons just as for a1) and for the di (by Corollary 3.2.12). This shows Theorem 3.2.2 and Corollary 3.2.4.

A direct application of Theorem 3.2.2 is as follows.

Corollary 3.2.17. Under the assumption that dim(Z) = 0, the ideal IX has a resolution of the following form:

0 Gn+1 Gn ... G2 G1 IX 0

i ∗ ∗ where Gi = ⊕ p Tij ⊗ O (−jξ) when i ∈ {1, . . . , n} and Gn+1 = p Tn ⊗ O (−ξ) j=1 P P for some locally free sheaves Tij and Tn over X.

3.2.3 Graded free resolution of the symmetric algebra Now, we turn to the analysis of a resolution of the symmetric algebra of a homogeneous ideal of the polynomial ring R = k[x0, . . . , xn]. So let IZ = (φ0, . . . , φn) ⊂ R be an ideal generated by n+1 linearly independent homogeneous polynomials each one of the same degree η ≥ 2. We will denote by RZ the quotient R/IZ and by Z n the subscheme V(IZ ) of P = Proj(R). We will assume that dim(Z) = 0 and that RZ is a graded Cohen-Macaulay ring. As above let:

M 0 Pn ... P2 P1 IZ 0 (P•) be a minimal graded free resolution of IZ , M being the presentation matrix of IZ n+1 n and P1 = R(−η) . The length of the resolution is equal to n = dim(P ) since Z is Cohen-Macaulay. 2 As in the previous section, let k1(Φ) : ∧ P1 → P1 be the second diﬀerential (φ0 ... φn) of the Koszul complex associated with the map Φ : P1 −−−−−−→ R. Put F = Im k1(Φ) in order to have the following diagram: 3.2. SUBREGULARITY OF THE SYMMETRIC ALGEBRA 73

k1(Φ) n+1 n+1 Φ R(−2η)( 2 ) R(−η) IZ 0. F 0 0

Deﬁnition 3.2.18. Set S = R[y0, . . . , yn] and y = (y0 . . . yn). We let IX be the ideal of S generated by the entries in the row matrix yM and IK be the ideal of S generated by the entries in the row matrix yk1(Φ).

Here, as above, F ⊂ E = Image(M) so IK ⊂ IX. Notation 3.2.19. Since S is bigraded by the variables x and y, S(−a, −b) stands for a shift in x for the left part and y for the right part. As above, we denote by P the product Pn × Pn and by p : Pn × Pn → Pn the ﬁrst projection.

To show Theorem 3.2.6, the strategy is initially the same as in the previous section, but since we are dealing with free resolutions, the resolutions of IK and IX/IK will patch together providing a resolution of IX without further checking. We will explain afterwards how we deduce from this resolution a minimal bigraded free resolution of IX.

The Koszul hull All the arguments of the proof of Proposition 3.2.11 remain valid in the graded homogeneous setting. So the ideal IK has the following properties:

(i) IK is a determinantal ideal. Under the assumption that codim(Z, Pn) = n:

(ii) codim(K, P) = n.

(iii) a graded free resolution of IK is the Eagon-Northcott complex associated to the matrix: φ . . . φ ψ = 0 n . y0 . . . yn

Hence, the following complex is a bigraded free resolution of IK:

0 Qn ... Q2 Q1 IK 0 (Q•)

i−1 where Qi = ⊕ Qi,j and j=0

(n+1) Qi,j = S − (i − j)η, −j − 1) i+1 for 1 ≤ i ≤ n and 0 ≤ j ≤ i − 1.

(iv) The scheme is Gorenstein and the canonical module ωS of veriﬁes: K K K ωS ' S n(η − 1) − 1, −n . K 74 CHAPTER 3. RESOLUTION OF THE SYMMETRIC ALGEBRA

Identiﬁcation of the quotient IX/IK

We denote by ωRZ the canonical module of Z. All the arguments of Proposi- tion 3.2.11 and Theorem 3.1.14 apply in the graded case since RZ is a graded Cohen-Macaulay ring of depth n. Hence we have that:

IX/IK ' ωRZ ⊗ S(n(1 − η) + 1, −1) as S-modules.

Recall that (P•) is a minimal graded free resolution of IZ . Put

0 ∨ Pi = Pn+1−i ⊗ S − nη, −1) for i ∈ {1, . . . , n + 1}. Then the complex:

Qn Q2 Q1 0 0 Pn+1 ⊕ ... ⊕ ⊕ IX 0 (R2’) 0 0 0 Pn P2 P1 is a bigraded free resolution of IX.

Homotopy of complexes

We turn now to the problem of extracting a minimal bigraded free resolution of IX from (R2’). In order to do so, we show ﬁrst the following result. Proposition 3.2.20. There is a canonical isomorphism

p∗OX(ξ) 'IZ where OX(ξ) and IZ are the sheaﬁﬁcation of respectively S(0, 1) and IZ . We emphasize that this is not completely straight forward since X is the Proj of IZ which is not locally free (see Stack project, 26.21. Projective bundles, example 26.21.2).

Proof. Since OP(ξ) is the relative ample line bundle of the projective bundle P = n+1 P OX (−η) , we have:  0 for l > 0 and j ≤ 0,  k 0 for j ∈ {1, . . . , k − 1} and any l, R p∗O (lη − jξ) = P O (lη) for k = 0 and j = 0,  X  n+1 OX (l − 1)η for k = 0 and j = −1.

Therefore, applying p∗ to the resolution (R1) and chasing cohomology we get 1 R p∗IX(ξ) = 0. n+1 Recall that we denote by E the kernel of Φ : OX (−η) → IZ and that IX(ξ) ∗ is the image of the map p E → OP(ξ). Let H be the kernel of this surjection and write the exact sequence:

∗ 0 H p E IX(ξ) 0. 3.2. SUBREGULARITY OF THE SYMMETRIC ALGEBRA 75

∗ 1 ∗ Since p∗p E'E and R p∗p E = 0, applying p∗ to this exact sequence, we get:

1 0 p∗H E p∗IX(ξ) R p∗H 0. (a)

1 Also, since we proved that R p∗IX(ξ) = 0, applying p∗ to the canonical exact sequence

0 IX(ξ) OP(ξ) OX(ξ) 0 we get

n+1 0 p∗IX(ξ) OX (−η) p∗OX(ξ) 0. (b)

The exact sequences (a) and (b) ﬁt into the following commutative diagram:

p∗H 0

E ' E

n+1 0 p∗IX(ξ) OX (−η) p∗OX(ξ) 0 = 1 0 R p∗H IZ p∗OX(ξ) 0

0 0

where (a) is the left column, (b) is the central row and the map IZ → p∗OX(ξ) in the bottom row is the canonical morphism associated to the projectivization of

IZ . This morphism is an isomorphism over X\Z and therefore IZ → p∗OX(ξ) is 1 injective because IZ is torsion free. Hence p∗H' 0 ' R p∗H and p∗OX(ξ) 'IZ .

Proof of Theorem 3.2.6. We work as in the previous proposition. Applying p∗ to the resolution of OX(ξ) given by (R1) and considering the associated R-modules of global sections, we obtain the following graded free resolution of IZ :

R − (n + 1)η ∨ ... 0 P0 − (n + 1)η ⊕ ∨ P0 − (n + 1)η

n+1 R(−2η)( 2 ) n+1 ⊕ R(−η) IZ 0. ∨ Pn (−(n + 1)η) 76 CHAPTER 3. RESOLUTION OF THE SYMMETRIC ALGEBRA

This resolution is homotopic to the minimal free resolution (P•) of IZ . There- fore, the truncated complex (P≥1) of (P•) is homotopic as a S-complex to:

Qn,0 Q1,0 0 0 Pn−1 ⊕ ... ⊕ . 0 0 Pn P1

Hence, (R2’) is homotopic to:

00 00 00 Qn−1 Qn−2 Q2 00 00 0 Qn ⊕ ⊕ ... ⊕ P1 IX 0 (R2) 00 00 00 Pn−1 Pn−2 P2 where

n n+1 00 ( i+1 ) 00 Qi = ⊕ Qi,j,Qi,j = S − (i − j)η, −j − 1) ,Pi = Pi+1 ⊗ S(η, −1). j=1

The complex (R2) is thus a bigraded free resolution of IX. To ﬁnish the proof of Theorem 3.2.6, it remains to show that (R2) is minimal. 0 This follows from the minimality of (P•) and the fact that, if i 6= i , there is no 00 00 00 bigraded homogeneous piece of the same degree among Qi and Qi0 or Pj for any j ∈ {1, . . . , n − 1}. Part II

Application to the study of rational maps

Chapter 4 n-to-n-tic maps

n n As we saw in Chapter2, given a rational map Φ : P 99K P defined by n + 1 homogeneous polynomials φ0, . . . , φn ∈ R = k[x0, ··· , xn] and denoting IZ = (φ0, . . . , φn) for the ideal sheaf of the base locus Z of Φ, we can study two schemes related to IZ . First, there is the graph Γ of Φ which is the Proj of the Rees algebra of IZ . By definition, the projective degrees of Φ are defined as the coefficients of the decomposition of Γ in cycle classes in Pn × Pn. Hence, we can read from Γ whether Φ is birational. Second, there is the projectivization X of the ideal of IZ which is the Proj of Sym(IZ ). Actually X contains Γ as an irreducible component and we have moreover the following commutative diagram:

Pn × Pn

p1 X p2

π1 π2 Γ σ1 σ2

n n P Φ P

n where p1 (resp. p2) is the projection over the ﬁrst factor P (resp. second factor Pn).

Notation. In the following, and with the notation in the previous diagram, we n denote by h1 (resp. h2) the class of a pull back of a hyperplane of P by p1 (resp. p2). Given a vector bundle G over Pn × Pn and two integers i and j, we denote also by G(i, j) the twisted vector bundle G(ih1 + jh2).

The equations of X and Γ in Pn × Pn are closely related to a free presentation of IZ . Namely, let

79 80 CHAPTER 4. n-TO-n-TIC MAPS

b φ0 . . . φn ⊕ O n (−i) i M n+1 P O n IZ (δ) 0 i≥1 P

E 0 0

n+1 be a locally free presentation of IZ and where E is the image of M in O n . Writing n n n P y0, . . . , yn for the variables of P , the equations of X in P × P are the entries of the row matrix y0 . . . yn M and we can compute the equations of Γ from these equations.

Deﬁnition 4.0.1. We call the sheaf E, the sheaf of relations of IZ .

Hence, if E is locally free, X is the zero locus of a global section of the vector ∗ ∨ n n bundle p1E (h2) over P × P . Moreover, if E is split as a sum of n line bundles n ∗ ∨ O n (−ai), p1E (h2) is equal to the sum ⊕ O n× n (aih1 + h2) and X is thus the P i=1 P P intersection of n divisors in Pn × Pn. Hence, when X has pure dimension n, it is a complete intersection in Pn × Pn and this last case makes particularly easy to compute the naive multidegree of Φ. As we will see in Definition 4.1.1 the naive multidegree of Φ is defined in the same way as the multidegree of Φ is defined, that is the multidegree (d0, . . . , dn) of Φ is the decomposition of Γ in cycle classes in n n CH(P × P ) and the naive multidegree (d0,..., dn) of Φ is the decomposition of X in cycle classes in CH(Pn × Pn). In the present chapter and Chapter5, we focus on two situations for which E is locally free. The first one is to assume that E is split. The second one will be to work on the base variety X = P2. Let focus on the first case. By the Hilbert-Burch theorem [Eis95, 20.15], in the case that Z has the expected codimension 2, the fact that E is split is equivalent to the fact that IZ is the ideal of maximal minors of M. Let us illustrate this with the following situation extracted from [DH17, Example 4.17].

Example 4.0.2. Let Φ be the rational map deﬁned by the 3 × 3-minors of the  2  −x1 x0 −x1 + x0x3 2  x0 x1 x0 − x1x2  matrix M =  . Hence the base ideal sheaf IZ has the  0 x2 x0x1 − x1x3  0 x3 −x0x1 + x0x2 following locally free presentation:

φ0 . . . φ3 2 M 4 0 O 3 (−1) ⊕ O 3 (−2) O 3 IZ (4) 0 P1 P1 P1

3 where O 3 is the structure sheaf of . P1 P1 3 3 Hence, the ideal I of X in P1 ×P2 is generated by the entries of the row matrix X y0 . . . y3 M, and since via a Macaulay2 computation we know that X has dimension n, we have that X is the complete intersection of two hypersurfaces of 4.1. EXPECTED DEGREES AND NAIVE PROJECTIVE DEGREES 81 bidegree (1, 1) and one hypersurface of bidedegree (2, 1). Hence the decomposition of X in cycle classes is

2 3 2 2 3 (h1 + h2) (2h1 + h2) = 2h1 + 5h1h2 + 4h1h2 + h2

∗ ∗ where h = c p O n (1) and h = c p O n (1) . Thus (d , d , d , d ) = 1 1 1 P 2 1 2 P2 0 1 2 3 (2, 5, 4, 1). Now, we focus on the decomposition of X into its irreducible components. As we explained in Proposition 2.2.13, the possible torsion components are ﬁbres over components of V Fitti(IZ ) for i ∈ {1, 2}. Here V Fitt2(IZ ) is empty but 3 Fitt1(IZ ) is supported over V(x0, x1) in P1. Actually, it is a computation, for example by computing the primary decomposition of X with Macaulay2, to show 3 3 that the ideal of the torsion component T in P × P is IT = (x0, x1, y2x2 + y3x3). To sum up X decomposes as the union of two components Γ and T each one of dimension 3. These two components have to be taken in account for the computation of the naive projective degrees of X. In this case, T is a complete intersection so its multidegree (1, 1, 0, 0) is easy to determine. Hence, the multidegree of Γ is (2−1, 5−1, 4, 1) = (1, 4, 4, 1). To explain this subtraction, we refer to the geometric interpretation of the naive multidegree or multidegree. For example, the nth naive 3 projective degree, is the coeﬃcient of h2 in the decomposition of X. But since T 3 3 intersect h2 once, Γ must intersect h2 once too.

By the previous analysis, we want to emphasize that the data of determinantal maps with given multidegree is equivalent to the data of the ideal of minors of the presentation matrix M. There are two main motivations for this chapter. First, as we mentioned in Theorem4, the multidegree of a Cremona map verifies the Cremona inequalities but it is not known if given a sequence (d0, . . . , dn) verifying Cremona inequalities, n n there exists a Cremona map P 99K P . Hence, the construction we propose is already a way to partially answer, even with n big, ProblemA concerning the Cremona inequalities. The second motivation concerns more precisely particular birational maps Φ : 3 3 −1 P1 99K P2 whose inverses Φ have the same algebraic degree than Φ. For simplicity we call these maps n-to-n-tics by analogy with the more common denomination cubo-cubic and quarto-quartic in degree 3 and 4. Recall from Subsection 1.3.3 that −1 the second multidegree d2 of Φ is the algebraic degree of Φ . Hence n-to-n-tic maps are the maps with multidegree (1, n, n, 1). Since quadro-quadrics (1, 2, 2, 1) rational maps of P3 cannot be defined by the maximal minors of a (4 × 3)-matrix of non constant entries, we did not focus on this case. Let us emphasize however that these quadro-quadric maps have been studied and classified via the XJC- correspondance (see [PR16]) in [PR13] and [PR14].

4.1 Expected degrees and naive projective degrees

n n With the previous notation, we let P 99K P be rational map. 82 CHAPTER 4. n-TO-n-TIC MAPS

Deﬁnition 4.1.1. We deﬁne the naive multidegree (d0,..., dn) of Φ as the multidegree of X of the base ideal sheaf I of Φ in Pn × Pn, i.e. n X i n−i [X] = dih1h2 i=0

th For i ∈ {0, . . . , n}, we call di the i naive projective degree. Let us illustrate this deﬁnition with the nth projective degree.

2 2 Example 4.1.2. Let Φ : P 99K P be a rational map. In this case the base locus Z of Φ has codimension 2 so, by 2.2.19, the possible torsion components of X have dimension 2 in P2 × P2 and X has pure dimension 2. 2 Since the graph Γ of Φ (resp. X) is deﬁned with two morphisms σ1 :Γ → P 2 2 2 and σ2 :Γ → P (resp. π1 : X → P and π2 : X → P ) which are the restriction to 2 2 2 2 2 2 Γ (resp. to X) of the two projections p1 : P × P → P and p2 : P × P → P , we summarize the situation with the following commutative diagram:

P2 × P2

p1 X p2

π1 π2 Γ σ1 σ2

2 2. P Φ P

2 2 In this case, c2 (O 2× 2 (0, 1) ) and c2 (O 2× 2 (0, 1) ) are both 0-cycles of P P X P P Γ P2 × P2 and the nth projective degree of Φ is the topological degree: 2 2 2 dt(Φ) = deg(c2 (OP ×P (0, 1) ) Γ) and the 2nd naive projective degree of Φ is:

2 d2 = deg(c2 (O 2× 2 (0, 1) ) ). P P X Let n ≥ 1 and let I be an ideal sheaf generated by n + 1 global sections of

n OP (δ), identiﬁed with homogeneous polynomials in n + 1 variables of degree δ. Assume that the sheaf of relations E of I is locally free, i.e. a locally free resolution of I reads n+1 0 E O n I(δ) 0. P Concerning this naive multidegree we have the following result. Proposition 4.1.3. The kth naive projective degree of Φ is equal to the degree of th ∨ ∨ n+1−k n the n + 1 − k Chern class cn+1−k(E ) of E (since CH (P ) ' Z). Proof. The kth projective degree of Φ is the degree of the support of the coker- n+1−k k nel W of a general morphism O n → I(δ). So we consider the following P commutative diagram: 4.1. EXPECTED DEGREES AND NAIVE PROJECTIVE DEGREES 83

k k O n = O n P P (φ . . . φ ) n+1 0 n 0 E O n I(δ) 0 P

n+1−k O n OW n+1−k 0 P

n+1−k n+1−k and we focus on the sequence E → O n → O n+1−k → 0. If W has the P W expected codimension n + 1 − k, then, by Proposition 1.2.6, the class [W n+1−k] is th ∨ ∨ equal to the n + 1 − k Chern class cn+1−k(E ) of E .

Proposition 4.1.4. Now suppose that the sheaf of relations E of I is split and n equal to ⊕ O n (−ai) for some integers ai ≥ 1. i=1 P Then the naive multidegree of the rational map Φ associated to I is

(dn,..., d0) = (sn(a1, . . . , an),..., s0(a1, . . . , an))

th where si(a1, . . . , an) is the i elementary function in a1, . . . , an.

Proof. This follows from applying Proposition 4.1.3 and computing the Chern classes of E.

Now, with the notation of the introduction to Chapter4, recall that the base ideal of a rational map Φ is of linear type if X = Γ. This implies that

(dn, . . . , d0) = (dn,..., d0).

n n Proposition 4.1.5. [DHS12, Corollary 2.6] Let Φ: P 99K P be a rational map whose base ideal IZ has a split sheaf of relation and that Z = V(IZ ) has codimension 2. Assume that I is of linear type, then Φ is birational if and only if the generators φ0, . . . , φn of I are the maximal minors of a n×n matrix M with only linear entries.

n Proof. Since E is split, put E = ⊕ O n (−ai) and write i=1 P n M φ0 . . . φn n n+1 ⊕ O (−ai) O n I(δ) 0 i=1 P P

P a locally free presentation of I (where δ = aii). As we explained in the intro- i duction, by the Hilbert-Burch theorem [Eis95, 20.15], this is equivalent to the fact that the φ0, . . . , φn are the n × n-minors of M (because Z has codimension 2). Now assuming that IZ is of linear type, of course if M has only linear entries i.e. ai = 1 for all i, dn = dn = sn(1,..., 1) = 1 so Φ is birational. Conversely, if Φ is birational, dn = dn = sn(a1, . . . , an) = 1 which implies that ai = 1 for all i since the ai are integers. So M has only linear entries.

We use the following notion of generality: 84 CHAPTER 4. n-TO-n-TIC MAPS

Deﬁnition 4.1.6. We say that a statement holds for a general matrix M if there 0 ∗ ∨ exists a Zariski dense subset U of H (p1 E )(h2) such that the statement holds ∗ ∗ for all p1M in U. If p1M lies in the complement of countably many Zariski closed 0 ∗ ∨ subsets of H (p1 E )(h2) , we say that M is very general.

4.2 Quarto-quartics

3 3 A determinantal rational map Φ : P 99K P of algebraic degree 4 is the data of its presentation (4 × 3)-matrix M necessarily with two columns of linear entries and one column with quadratic entries, i.e. a locally free resolution of the base ideal sheaf of IZ reads: φ0 . . . φ3 2 M 4 0 E = O 3 (−1) ⊕ O 3 (−2) O 3 IZ (4) 0. P1 P1 P1

∗ 0 ∗ ∨ Proposition 4.2.1. If p1M is general in H p1(E )(h2) , then the multidegree and naive multidegree of Φ are the same and are equal to (2, 5, 4, 1).

∗ ∨ Proof. First, this is a theorem of Bertini [Har77, 7.9.1] (p1(E )(h2) being very 0 ∗ ∨ ample since all the ai are strictly positive) that a section of H (p1(E )(h2)) is smooth and irreducible so the projectivization X of IZ is equal to the blow-up of IZ . In other words, IZ is of linear type so the multidegree and the naive multidegree of Φ coincide. Now the symmetric functions in 1, 1, 2 give the multidegree (2, 5, 4, 1). As we saw, in order to create a torsion component which decreases the 3rd and 2nd projective degree by one, we can make the matrix M have rank 1 over a curve 3 in P1. Since some of the entries of M are linear this curve is necessarily a line, for instance the line V(x0, x1). We construct a quarto-quartic map as follows. Let

 1 1 1 1  a0x0 + a1x1 + a2x2 + a3x3 2 2 2 2 a0x0 + a1x1 + a2x2 + a3x3 M1 =  3 3 3 3  a0x0 + a1x1 + a2x2 + a3x3 4 4 4 4 a0x0 + a1x1 + a2x2 + a3x3  1 1  b0x0 + b1x1 2 2 b0x0 + b1x1 M2 =  3 3  b0x0 + b1x1 4 4 b0x0 + b1x1  1 2 1 1 1 1 2 1 1  c2000x0 + c1100x0x1 + c1010x0x2 + c1001x0x3 + c0200x1 + c0110x1x2 + c0101x1x3 2 2 2 2 2 2 2 2 2 c2000x0 + c1100x0x1 + c1010x0x2 + c1001x0x3 + c0200x1 + c0110x1x2 + c0101x1x3 M3 =  3 2 3 3 3 3 2 3 3  c2000x0 + c1100x0x1 + c1010x0x2 + c1001x0x3 + c0200x1 + c0110x1x2 + c0101x1x3 4 2 4 4 4 4 2 4 4 c2000x0 + c1100x0x1 + c1010x0x2 + c1001x0x3 + c0200x1 + c0110x1x2 + c0101x1x3

i i be the respective ﬁrst, second and third columns of M where the coeﬃcients aj, bj i and cklmn are in k, and assume that the collection of those polynomials is general. Proposition 4.2.2. Given that M is general among the conditions we imposed, ∗ i.e. p1M is general in

0 ∗ H p O 3 (1) ⊕ I (1) ⊕ I (2) (h ) , 1 P V(x0,x1) V(x0,x1) 2 4.2. QUARTO-QUARTICS 85 the determinantal map of the 3 × 3 minors of M has multidegree (1, 4, 4, 1).

∗ 3 3 Proof. To define M, put L = V(x0, x1) and let L1 = p1(L) in P ×P . Our definition 0 3 3 of the generality of M is the same as considering s1 general in H OP ×P (1, 1) , 0 0 s2 general in H IL1 (1, 1) , s3 general in H IL1 (2, 1) and considering that X is the intersection V(s1) ∩ V(s2) ∩ V(s3). But this subvariety is smooth outside L1 by a theorem of Bertini [Har77, 7.9.1] so, by analysing the Fitting ideals of M, for instance Fitt1(M) is supported over V(x0, x1) and the other Fitting ideals are zero, it has two components. One is necessarily the blow-up Γ of I. We denote by T the other one supported over L1. This latter component T is reduced. Indeed, by Bertini’s theorem, we can show this property of being reduced by studying 0 the intersection V(s2) ∩ V(s3) for s2 general in H IL1 (1, 1) and s3 general in 0 H IL1 (2, 1) and showing that it is reduced. Since being reduced is an open property, it suffices to show the reduction for a specific choice of s2 and s3. So let

3 3 V(s2) = (H1 × P ) ∪ (P × H2)

3 where H1 is a hyperplane of the ﬁrst factor P containing L and

3 3 0 V(s3) = (C1 × P ) ∪ (P × H2)

3 where C1 is a quadric surface in the ﬁrst factor P containing L. Then, in particular 3 since the intersection of H1 and C1 in P is equal to the (reduced) union of L with another line L0, we have

3 0 3 0 3 0 V(s2) ∩ V(s3) = (L × P ) ∪ (L × P ) ∪ (H1 × H2) ∪ (C1 × H2) ∪ P × (H2 ∩ H2) which is reduced. Hence T is reduced. So, since T coincides set-theoretically with P(I(4) L), we have T = P(I(4) L). 2 Actually I(4) L 'OL ⊕ OL(1) by restriction of the presentation matrix M of 2 I over L where the columns of M vanish. Hence P I(4) L is a P -fibration over 2 3 L so that the coefficients of h1h2 and h2 are respectively 1 and 1. To check the 2 second coefficient, note that P(I(4) L) = P OL ⊕ OL(1) and that the map given 3 3 by the relatively ample line bundle is the blow-up of a line in P . So the cycle h1T is given by the inverse image of a general point of P3 under this blow up, and is thus a single point. 3 For the first coefficient, restricting I(4) L to a general plane P we get Ox where 3 x = L ∩ P . So the cycle P(I(4) L)h1 is the plane P(Ox), which maps to a plane in P3 and hence cuts a general line along a single point. To sum up, T is the intersection of L1 with a general divisor of class (1, 1) in 3 3 2 2 P × P , so L1 = h1 and T = h1(h1 + h2). So the multidegree of T is (1, 1, 0, 0) and the multidegree of Γ is necessarily (1, 4, 4, 1).

In the article [DH17], J.D´esertiand F.Han provide a description of the base locus of such a general quarto-quartic map.

3 3 Proposition 4.2.3. Let Φ: P1 99K P2 be a quarto-quartic map whose presentation ∗ matrix M veriﬁes that p1M is general in

0 ∗ H p O 3 (1) ⊕ I (1) ⊕ I (2) (h ) . 1 P V(x0,x1) V(x0,x1) 2 86 CHAPTER 4. n-TO-n-TIC MAPS

Then the base locus Z of Φ is the union of the line L = V(x0, x1) of multiplicity 3 and a smooth irreducible curve C of genus 5 and degree 8. Moreover L is 5-secant to C. Let us emphasize that the total degree 11 of the union of the line L and C is coherent with the fact that the curve deﬁned by the ideal I3(M) of 3 × 3-minors of ∗ 0 ∗ 2 3 3 a matrix M such that p1M is general in H (p1(OP (1) ⊕ OP (2))(h2)) is a smooth curve of degree 11. Indeed, consider M as a matrix with entry in the polynomial ring R = k[x0, x1, x2, x3] and consider the ideal I3(M) of 3 × 3-minors of M. By the genericity of M, the Eagon-Northcott complex associated to M provided a minimal free resolution of R/I3(M). Namely,

2 M 4 0 O 3 (−5) ⊕ O 3 (−6) R(−4) R R/I3(M) 0 (4.2.1) P1 P1 is a minimal free resolution of R/I3(M). Now, as it is explained in [Sch03, 3.2 Free resolutions, page 46], we can compute the Hilbert polynomial of I3(M) via this resolution, it is equal to HP (I3(M),T ) = 11T − 13 so the degree of the curve 3 deﬁned by I3(M) in P is equal to 11. Proposition 4.2.4. [DH17, Remark 4.6] The inverse of a determinantal quarto- quartic map as in Proposition 4.2.3 is a determinantal quarto-quartic map. Remark 4.2.5. Let us explain how computationally (empirically) we recover this 3 3 fact. Let Φ : P1 99K P2 be a determinantal quarto-quartic general map for which we denote by φ0 . . . φ3 2 M 4 0 O 3 (−1) ⊕ O 3 (−2) O 3 IZ (4) 0. P1 P1 P1 a presentation of IZ . We can assume without lost of generality that the line supporting the ideal sheaf Fitt0(IZ ) is the line V(x0, x1) and that M is given as in 3 3 the proof of Proposition 4.2.3. The ideal IX of the projectivization of X in P1 × P2 is generated by two sections of bidegree (1, 1) and one section of bidegree (2, 1) (the entries of the row matrix y0 y1 y2 y3 M). Then it is a computation to show that the ideal IΓ of the graph Γ of Φ, that is ∞ the saturation [IX :(x0, x1) ] of IX by (x0, x1) (where we consider (x0, x1) as an 3 3 ideal sheaf over P1 ×P2) is generated by two sections of bi-degree (1, 1), one section 0 of bi-degree (2, 1) and one section of bi-degree (1, 2). Now the projectivization IX 0 −1 of the projectivization X of the base ideal IZ0 of Φ should be generated by the two sections of bidegree (1, 1) and the section of bidegree (1, 2) generating IΓ. Indeed, X0 contains the graph Γ of Φ (because it is also the graph of Φ−1) as an irreducible component and is generated by at least three sections of bi-degree (1, ∗) because Γ has codimension 3. In this case, these three sections of bi-degree (1, ∗) are indeed the two sections of bidegree (1, 1) and the section of bidegree (1, 2)

0 generating IΓ. Since the ideal of the projectivization IX of the projectivization 0 −1 X of the base ideal IZ0 of Φ is generated by two sections of bidegree (1, 1) and one section of bidegree (1, 2), the presentation matrix of IZ0 has two columns with linear entries and one column of quadratic entries and Φ−1 is thus a determinantal (quarto-quartic) map. See Chapter7 for the details of the last argument. 4.2. QUARTO-QUARTICS 87

One way to complete our proof that X0 is indeed generated by the two sections of bidegree (1, 1) and the section of bidegree (1, 2) generating IΓ would be to show that Γ = X ∩ X0 but, even if we could verify this fact in all our examples, we could not provide of a complete proof of it.

Remark 4.2.6. We emphasize that given the line V(x0, x1), we can consider the matrix  1 1 1  λ01 l01 q0123 2 2 2 λ01 l01 q0123 M =  3 3 3  λ01 l01 q0123 4 4 4 λ01 l01 q0123 i i where the entries l01 and λ01 are linear polynomials in the variables x0, x1 and the i entries q0123 are quadratic polynomials in the variables x0, x1, x2, x3. Let Φ be the determinantal map of the 3×3 minors of M. In this case, the torsion component T 3 P i+1 3 3 of X has ideal IT = (x0, x1, yiq0123) and hence its decomposition in CH(P1 ×P2) i=0 2 is h1(2h1 +h2) and its multidegree is thus (2, 1, 0, 0) so the multidegree of the graph Γ is (0, 4, 4, 1) and Φ is not dominant. This is why we only consider the case that M has one of its column with linear entries and the columns of quadratic entries which vanish over the line V(x0, x1). Now, we can make an enumeration of the parameters of the determinantal quarto-quartics. As we see, up to a choice of coordinate, a determinantal quarto- quartic is the data of a matrix M such that  1 1 1 1  a0x0 + a1x1 + a2x2 + a3x3 2 2 2 2 a0x0 + a1x1 + a2x2 + a3x3 M1 =  3 3 3 3  a0x0 + a1x1 + a2x2 + a3x3 4 4 4 4 a0x0 + a1x1 + a2x2 + a3x3  1 1  b0x0 + b1x1 2 2 b0x0 + b1x1 M2 =  3 3  b0x0 + b1x1 4 4 b0x0 + b1x1  1 2 1 1 1 1 2 1 1  c2000x0 + c1100x0x1 + c1010x0x2 + c1001x0x3 + c0200x1 + c0110x1x2 + c0101x1x3 2 2 2 2 2 2 2 2 2 c2000x0 + c1100x0x1 + c1010x0x2 + c1001x0x3 + c0200x1 + c0110x1x2 + c0101x1x3 M3 =  3 2 3 3 3 3 2 3 3  c2000x0 + c1100x0x1 + c1010x0x2 + c1001x0x3 + c0200x1 + c0110x1x2 + c0101x1x3 4 2 4 4 4 4 2 4 4 c2000x0 + c1100x0x1 + c1010x0x2 + c1001x0x3 + c0200x1 + c0110x1x2 + c0101x1x3 are the respective ﬁrst, second and third columns of M. This a total of 52 parameters. However, diﬀerent choices of parameters can produce the same map Φ. This is the case for example if we compose a matrix G ∈ Gl4(k) with M. This should give a way to recover the following result Proposition 4.2.7. [DH17, Proposition 4.8] The family of determinantal quarto- quartic has dimension 46 in the Cremona group Bir(P3, P3). However, we did not have time to push that far our results.

Determinantal quinto-quartic interlude With our point of view of enumerating the torsion components of the projectivization of the base ideal IZ of a determinantal quartic map, we can consider the case 88 CHAPTER 4. n-TO-n-TIC MAPS of quinto-quartics, that is maps of P3 of multidegree (1, 5, 4, 1). The resolution of IZ still reads φ0 . . . φ3 2 M 4 0 O 3 (−1) ⊕ O 3 (−2) O 3 IZ (4) 0. P1 P1 P1 and Φ still has naive multidegree (d3, d2, d1, d0) = (2, 5, 4, 1). Hence, here we are looking for a torsion component of dimension 3 inﬂuencing d3 but not d2, that is 3 3 3 torsion component with cohomological decomposition h1 in CH(P1 × P2). Those possible torsion components are thus located above a closed point of P3. So let 3 choose the point V(x0, x1, x2). Since it is of codimension 3 in P , it has to be the support of Fitt2(IZ ) i.e. support of the ideal of 1 × 1 minors of M. For instance, let  1 1 1  λ012 l012 q012 2 2 2 λ012 l012 q012 M =  3 3 3  λ012 l012 q012 4 4 4 λ012 l012 q012 i i where the entries l012 and λ012 are linear polynomials in the variables x0, x1, x2 i and the entries q012 are quadratic polynomials whose monomials are all divisible by the variables x0, x1, x2 but else general in those conditions. The map of 3 × 3 minors of M has then multidegree (1, 5, 4, 1).

3 ∗ Proposition 4.2.8. Let z be a point in P and let z1 = p1z. Now let M be a ∗ matrix such that p1M is very general in

0 2 H Iz1 (1, 1) ⊕ Iz1 (2, 1) .

Then the determinantal map Φ associated to the 3×3-minors of M has multidegree (1, 5, 4, 1) i.e. is a quinto-quartic.

Proof. In the same way as in the proof of Proposition 4.2.2, the projectivization X of the base ideal sheaf I of Φ has two reduced components. One is the blow- up Γ of Φ, the other is V(z1) which has multidegree (1, 0, 0, 0). Hence, since the multidegree of X is (2, 5, 4, 1), the multidegree of Γ is (1, 5, 4, 1) which is thus the multidegree of Φ.

Computationally we observe that the base locus Z of a general determinantal quinto-quartic is a curve of degree 11 and arithmetic genus 14, singular at the support of the torsion component. Its inverse is a determinantal quintic whose base locus Z0 is the union of two singular curve of degree 9, one of arithmetic genus 7, the other of arithmetic genus 8. We give the computation with Macaulay2. We make the computation over Z/5Z because it is much simpler. In order to impose the general conditions on polynomials whose monomials are divisible by x0, x1, x2 we give a weight to each variable x0, x1, x2, x3 at ﬁrst. i1 : k = ZZ/5 o1 = k 4.2. QUARTO-QUARTICS 89 o1 : QuotientRing i2 : R = k[x_0,x_1,x_2,x_3,Degrees=>{{1,0,0,0},{0,1,0,0}, {0,0,1,0},{0,0,0,1}}] o2 = R o2 : PolynomialRing

We then deﬁne the polynomials in the matrix M. i3 : Q1 = random({2,0,0,0},R)+random({1,1,0,0},R)+ random({1,0,1,0},R)+random({1,0,0,1},R)+random({0,2,0,0},R)+ random({0,1,1,0},R)+random({0,1,0,1},R)+random({0,0,2,0},R)+ random({0,0,1,1},R)

2 2 2 o3 = x - 2x x - x - x x + 2x x - 2x + x x 0 01 1 02 12 2 03 o3 : R i4 : Q2 = random({2,0,0,0},R)+random({1,1,0,0},R)+ random({1,0,1,0},R)+random({1,0,0,1},R)+random({0,2,0,0},R)+ random({0,1,1,0},R)+random({0,1,0,1},R)+random({0,0,2,0},R)+ random({0,0,1,1},R)

2 2 o4 = - x - 2x x + x + 2x x + 2x x + x x 0 0 1 1 1 2 0 3 2 3 o4 : R i5 : Q3 = random({2,0,0,0},R)+random({1,1,0,0},R)+ random({1,0,1,0},R)+random({1,0,0,1},R)+random({0,2,0,0},R)+ random({0,1,1,0},R)+random({0,1,0,1},R)+random({0,0,2,0},R)+ random({0,0,1,1},R)

2 2 2 o5 = - x + 2x - x x + 2x x + x - 2x x + 2x x 0 1 02 12 2 03 23 o5 : R 90 CHAPTER 4. n-TO-n-TIC MAPS i6 : Q4 = random({2,0,0,0},R)+random({1,1,0,0},R)+ random({1,0,1,0},R)+random({1,0,0,1},R)+random({0,2,0,0},R)+ random({0,1,1,0},R)+random({0,1,0,1},R)+random({0,0,2,0},R)+ random({0,0,1,1},R)

2 2 2 o6 = x - x x - 2x - x x - 2x x - x + x x 0 01 1 02 12 2 23 o6 : R i7 : L1 = random({1,0,0,0},R)+random({0,1,0,0},R)+ random({0,0,1,0},R) o7 = -x 0 o7 : R i8 : L2 = random({1,0,0,0},R)+random({0,1,0,0},R)+ random({0,0,1,0},R) o8 = 2x + 2x - x 0 1 2 o8 : R i9 : L3 = random({1,0,0,0},R)+random({0,1,0,0},R)+ random({0,0,1,0},R) o9 = 2x + 2x 0 1 o9 : R i10 : L4 = random({1,0,0,0},R)+random({0,1,0,0},R)+ random({0,0,1,0},R) o10 = 2x - 2x 1 2 o10 : R i11 : K1 = random({1,0,0,0},R)+random({0,1,0,0},R)+ random({0,0,1,0},R) o11 = x + x + x 0 1 2 4.2. QUARTO-QUARTICS 91

o11 : R i12 : K2 = random({1,0,0,0},R)+random({0,1,0,0},R)+ random({0,0,1,0},R) o12 = -x 1 o12 : R i13 : K3 = random({1,0,0,0},R)+random({0,1,0,0},R)+ random({0,0,1,0},R) o13 = 2x - 2x + x 0 1 2 o13 : R i14 : K4 = random({1,0,0,0},R)+random({0,1,0,0},R)+ random({0,0,1,0},R) o14 = 2x - 2x + 2x 0 1 2 o14 : R i15 : M = matrix{{Q1,Q2,Q3,Q4},{L1,L2,L3,L4},{K1,K2,K3,K4}} o15 = | x_0^2-2x_0x_1-x_1^2-x_0x_2+2x_1x_2-2x_2^2+x_0x_3 | -x_0 | x_0+x_1+x_2 ------x_0^2-2x_0x_1+x_1^2+2x_1x_2+2x_0x_3+x_2x_3 2x_0+2x_1-x_2 -x_1 ------x_0^2+2x_1^2-x_0x_2+2x_1x_2+x_2^2-2x_0x_3+2x_2x_3 2x_0+2x_1 2x_0-2x_1+x_2 ------x_0^2-x_0x_1-2x_1^2-x_0x_2-2x_1x_2-x_2^2+x_2x_3 | 2x_1-2x_2 | 2x_0-2x_1+2x_2 |

3 4 o15 : Matrix R <--- R 92 CHAPTER 4. n-TO-n-TIC MAPS i16 : T = k[x_0,x_1,x_2,x_3] o16 = T o16 : PolynomialRing i17 : M = sub(M,T); i18 : I = minors(3,M); i19 : degree I,genus I o19 = (11, 14) o19 : Sequence i20 : time primaryDecomposition I -- used 0.131974 seconds

We do not print the output corresponding to the command primaryDecomposition of I but the result is that V(I) is irreducible because there is one element in the resulting list. i21 : time radical (primaryDecomposition ideal singularLocus I)_0 -- used 1.13423 seconds o21 = ideal (x , x , x ) 2 1 0 o21 : Ideal of T i23 : loadPackage "Cremona"; o23 = Cremona o23 : Package i24 : psi = toMap(minors(3,M)); o24 : RingMap T <--- T i28 : projectiveDegrees psi o28 = {1, 4, 5, 1} 4.2. QUARTO-QUARTICS 93 o28 : List i30 : phi = inverseMap psi; o30 : RingMap T <--- T i31 : idPhi = ideal( phi(x_0) , phi(x_1) , phi(x_2) , phi(x_3)); i32 : betti res idPhi

0 1 2 o32 = total: 1 4 3 0: 1 . . 1: . . . 2: . . . 3: . . . 4: . 4 2 5: . . . 6: . . 1 o32 : BettiTally

Remark 4.2.9. Let us remark that we could determine the multidegree of Φ without using the Cremona package. Indeed, the torsion component T is supported over V(x0, x1, x2) so we can saturate X in order to recover the blow-up of I. This is the content of the following code. We start the previous code at the line o21. i22 : S = T[y_0,y_1,y_2,y_3] o22 = S o22 : PolynomialRing i23 : J = ideal( matrix{{y_0,y_1,y_2,y_3}}*sub(transpose M,S)); o23 : Ideal of S i24 : Graph = saturate(J,sub(ideal(x_0,x_1,x_2),S)); o24 : Ideal of S Via the generator of the ideal Graph we can then compute the kth projective 3 3 3−k k degree dk of Φ by intersecting Γ with the plane of P × P of class h1 h2 . We provide one command for the computation of d3 (the other projective degree can be computed in a similar way). i25 : H3 = ideal (matrix{{y_0,y_1,y_2,y_3}}* random(S^{4:{1,1}},S^{3:{1,1}})); 94 CHAPTER 4. n-TO-n-TIC MAPS

o25 : Ideal of S i26 : F3 = Graph+H3; o26 : Ideal of S i27 : PDF3 = primaryDecomposition F3 o27 = {ideal (x - x , x - x , x + x , y , y - 2y , y + y ), 23130331 202 ------3 ideal (y , y , y , y ), ideal (x , x , x , y , y - 2y , y + y , y )} 3210 21031 2022 o27 : List i28 : degree PDF3_0 o28 = 1

traducing the fact that d3 = 1 and that Φ is birational.

4.3 Determinantal quinto-quintics

We turn now to determinantal quinto-quintics. We emphasize that there is two partitions of the number 5 in three positive integers, namely 5 = 1 + 1 + 3 and 5 = 1 + 2 + 2. Hence the resolution of the base ideal sheaf IZ of a determinantal quinto-quintic is one of the followings:

2 M 4 (1) 0 O 3 (−1) ⊕ O 3 (−3) O 3 IZ (5) 0 P1 P1 P1

2 M 4 (2) 0 O 3 (−1) ⊕ O 3 (−2) O 3 IZ (5) 0 P1 P1 P1

We propose constructions in both cases.

(1) We focus ﬁrst on the case that the base ideal sheaf IZ of a determinantal 3 3 quinto-quintic Φ : P1 99K P2 has resolution

2 M 4 0 O 3 (−1) ⊕ O 3 (−3) O 3 IZ (5) 0 P1 P1 P1

We denote by X the projectivization of IZ . It is the complete intersection of two divisors of bidegree (1, 1) and one divisor of bidegree (3, 1). Hence its 4.3. DETERMINANTAL QUINTO-QUINTICS 95

3 3 decomposition in CH(P1 × P2) is: 2 3 2 2 3 [X] = (h1 + h2) (3h1 + h2) = 3h1 + 7h1h2 + 5h1h2 + h2.

Thus the naive multidegree (d0, d1, d2, d3) of Φ is equal to (3, 7, 5, 1) i.e. if M has a general collection of entries, Φ has multidegree (3, 7, 5, 1) (this follows also from Proposition 4.1.4). Hence if Φ has multidegree (1, 5, 5, 1), X must present torsion components whose multidegree adds up to (2, 2, 0, 0) which is 3 2 to say, torsion components whose total class is 2h1 + 2h1h2. We propose two constructions for this multidegree.

(a) The ﬁrst solution is to construct two torsion components each one of mu- 3 tidegree (1, 1, 0, 0). So we choose two lines in P1, for instance V(x0, x1) and V(x2, x3) and we let  1 1 1 λ01 l23 c 2 2 2 λ01 l23 c  M =  3 3 3 λ01 l23 c  4 4 4 λ01 l23 c i i where for i ∈ {1,..., 4}, the entries λ01 (resp. l23) are linear homo- i geneous polynomial in the variables x0, x1 (resp. x2, x3) and c is a homogeneous polynomial of degree 3 with all its monomials divisible by x0x2 or x0x3 or x1x2 or x1x3. Proposition 4.3.1. Let M be the previous matrix and assume that it is general among the conditions we imposed. Then the determinantal map of the 3 × 3 minors of Φ is a quinto-quintic. We call such a map a quinto-quintic of type (a).

Proof. Since the entries of M are general among the conditions we imposed, we have that Fitt1(IZ ) is supported over the union V(x0, x1) ∪ V(x2, x3). Hence the two torsion components over the lines V(x0, x1) 2 3 and V(x2, x3) have dimension 3 and decomposition h1(h1 + h2) = h1 + 2 3 3 h1h2 in CH(P1 × P2), we refer to Remark 4.2.6 for the details of this argument. Hence, since there is no other torsion components, the component Γ of the graph has multidegree (3 − 2, 7 − 2, 5, 1) = (1, 5, 5, 1) i.e. Φ is a quinto-quintic. Example 4.3.2. An example of quinto-quintic of type (a) is the the determinantal map of the 3 × 3 minors of the following presentation matrix:  2 2 x0 x2 + x3 x0x2 + x0x1x2 + x0x3 2 3x0 + x1 x2 + 2x3 x1x3 + x1x2x3  M =  2   x0 + x1 x2 x1x2 + x0x1x3  2 2 x0 + 2x1 x3 x0x3 + x1x3

The base locus Z of Φ has degree 18 and is the union of the two lines V(x0, x1) and V(x2, x3) and an irreducible smooth curve of degree 12 and genus 9. Moreover this latter curve is 8-secant to V(x0, x1) and 8-secant V(x2, x3). 96 CHAPTER 4. n-TO-n-TIC MAPS

Remark 4.3.3. In all the examples of quinto-quintic of type (a) we have considered, the inverse was also of type (a) but we could not establish this fact in all generality.

(b) We turn now to the construction of a single torsion component of multidegree (2, 2, 0, 0). As we said in the introduction, for the moment, we only control the reduced structure of X which, concerning the location of the torsion components, is computed by the ﬁtting ideal of the base ideal sheaf IZ . Hence, our strategy is to construct a torsion component with big enough multidegree supported over a line, for example V(x0, x1). For instance, consider the matrix  1 1 1  l0123 l01 c01 2 2 2 l0123 l01 c01 M =  3 3 3  l0123 l01 c01 4 4 4 l0123 l01 c01 i where for all i ∈ {1,..., 4} l0123 is a linear homogeneous polynomial i in the variables x0, x1, x2, x3, l01 is a linear homogeneous polynomial in i the variables x0, x1 and c01 is a cubic homogeneous polynomial whose 2 2 monomials are divisible by x0 or x0x1 or x1. If the entries of M are general under those conditions, our conjecture is that the associated determinantal map is a quinto-quintic. We call determinantal quinto- quintic of type (b). An example of such a quinto-quintic is as follows, the computation showing this result and the properties of the base locus was made with Macaulay2. Proposition 4.3.4. Let  3 2 3 2  x0 x0 + x2 + x3 x0 + x0x1 + x1 + 2x0x2 3 2  x1 x1 + x3 2x1 + x0x2 + x0x1x3  M =  2 2  . x0 + x1 x2 + x3 2x0x2 + x0x1x2 + x0x3  3 2 2 x0 x2 + x3 x1 + 2x0x1x2 + x1x2 + x1x3 Then the determinantal map associated to 3×3 minors of M is a quinto- quintic (of type b).

Its base locus ideal IZ has degree 18 and is the union of a smooth irreducible curve of genus 8 and degree 11 and the line V(x0, x1). Moreover the curve is 8 secant to V(x0, x1). The inverse of Φ is also a quinto- quintic of type (b).

(2) We turn now to the case that the base ideal sheaf IZ of a determinantal 3 3 quinto-quintic Φ : P 99K P has resolution

2 M 4 0 O 3 (−1) ⊕ O 3 (−2) O 3 IZ (5) 0. P P P

0 2 3 3 3 If M is general in H (OP (1) ⊕ OP (2) ), the decomposition of X in CH(P × P3) is: 2 3 2 2 3 [X] = (h1 + h2)(2h1 + h2) = 4h1 + 8h1h2 + 5h1h2 + h2 4.4. HIGHER DEGREE AND HIGHER DIMENSION 97

∗ ∗ where h = c p O n (1) and h = c p O n (1) so 1 1 1 P 2 1 2 P2

(d3, d2, d1, d0) = (4, 8, 5, 1).

In this case, our strategy is to construct one torsion component with dimension strictly bigger than n, for instance, by providing the vanishing of M over the line V(x0, x1). Here, X is no more a complete intersection so the naive multidegree of the associated map is not (4, 8, 5, 1) anymore. Hence, consider the matrix

 1 1 1  l01 k01 q01 2 2 2 l01 k01 q01 M =  3 3 3  l01 k01 q01 4 4 4 l01 k01 q01

i where for all i ∈ {1,..., 4} l01 is a linear homogeneous polynomial in the i i variables x0, x1 and k01 and q01 are quadric homogeneous polynomials whose monomial are divisible by x0 or x1. We observe that if the entries of M are general among those conditions, then the determinantal map of 3 × 3 minors of M is a quinto-quintic whose inverse is a determinantal quinto-quintic whose projectivization of base locus X0 has also a single torsion component supported over a line. An example of such a quinto-quintic is as follows, the computation showing this result and the properties of the base locus was made with Macaulay2.

Proposition 4.3.5. Let

 2 2  2x0 + x1 2x1 + x0x3 x0 + x0x3 2  x0 x1x2 + x1x3 x0 + x0x1 + x1x2 M =  2 2  .  x0 + x1 x0 + x0x2 x0x1 + x1  2 2 2 x0 x0 + x1 x0 + x1x2 + x1x3

Then the determinantal map associated to the 3 × 3 minors of M is a quinto- quintic.

Its base locus ideal IZ has degree 17 and is the union of a smooth irreducible curve of genus 8 and degree 11 and the line V(x0, x1). Moreover the curve is 8-secant to V(x0, x1). The inverse of Φ is also a quinto-quintic of of the same type.

4.4 Higher degree and higher dimension

To ﬁnish this section, we propose now a summary of our investigation until now. Instead of writing explicitly the conditions under which the entries of the matrix M has to be taken general, we write an example of matrix whose associated determinantal map should represent the general properties of the family. 98 CHAPTER 4. n-TO-n-TIC MAPS

3 4.4.1 Some families of P

Multidegree Example Base locus Z (1, 4, 4, 1)

(112)

 2  Z has degree 11 and is the union −x1 x0 −x1 + x0x3 2  x0 x1 x − x1x2  of a smooth irreducible curve C of  0   0 x2 x0x1 − x1x3  degree 8 and genus 5 and a line l. 0 x3 −x0x1 + x0x2 The curve C is 5-secant to l. (1, 5, 4, 1)

(112)  2  x2 x1 x0 + x0x3 Z is an irreducible curve C of de- 2 x0 + x2 x0 + x2 x + x1x3   0  gree 11 and arithmetic genus 14 x0 + x1 0 x1x3 + x2x3 2 singular in one point. 0 x2 x0 + x1x2 (1, 5, 5, 1)

(113(a)) Z has degree 18 and is the union  2 2 x0 x2 + x3 x0x2 + x0x1x2 + x0x3 of a smooth irreducible curve C of 2 3x0 + x1 x2 + 2x3 x1x3 + x1x2x3   2  degree 12 and genus 9 and two lines  x0 + x1 x2 x1x2 + x0x1x3  2 2 l1 and l2. The curve C is 8-secant x0 + 2x1 x3 x0x3 + x1x3 to l1 and 8-secant to l2.

(113(b))

 2 2 Z has degree 18 and is the union x0 x2 + x3 x0x2 + x0x1x2 + x0x3 2 3x0 + x1 x2 + 2x3 x1x3 + x1x2x3  of a smooth irreducible curve C of  2   x0 + x1 x2 x1x2 + x0x1x3  degree 11 and genus 8 and one line x + 2x x x2x + x x2 0 1 3 0 3 1 3 l. The curve C is 8-secant to l.

(122)

Z has degree 17 and is the union  2 2  2x0 + x1 2x1 + x0x3 x0 + x0x3 of a smooth irreducible curve C of 2  x0 x1x2 + x1x3 x0 + x0x1 + x1x2  2 2  degree 11 and genus 8 and one line  x0 + x1 x0 + x0x2 x0x1 + x1  2 2 2 l. The curve C is 8-secant to l1 and x0 x0 + x1 x0 + x1x2 + x1x3 8-secant to l2.

We emphasize that in the example of quinto-quintic of type (122), the projectivization X has a torsion component of dimension 4 above the line V(x0, x1). 4.4. HIGHER DEGREE AND HIGHER DIMENSION 99

Multidegree Example Base locus Z (1, 7, 5, 1)

(113)  3 3 2 3  x0 + x2 x2 x0 + x1 + x0x2 + x2 + x0x1x3 Z is a singular irreducible 2 3 2 2 x0 + x1 x1 + x2 x0x1 + x1 + x0x2 + x2x3   2 2  curve of degree 18 and arith-  x2 x0 x0x2 + x0x1x2 + x1x2 + x1x2x3 2 2 2 metic genus 39. x0 + x1 x2 x0x1 + x0x3 + x1x3 + x1x2x3 (1, 6, 6, 1)

(114(a)) Z has degree 27 and is the u- nion of a smooth irreducible  2 2 2 2 2  x0 x2 x1x2 + x0x2x3 + x0x3 curve C of degree 17 and genus 2 2 2 3  x1 x2 x0x1x2 + x0x2 + x0x3 + x0x1x2x3  2 3 2 2  15 with one lines l1 of degree x0 + x1 x2 − x3 x0x1x2 + x1x2 + x0x1x3 + x0x2x3 2 3 2 2 2 7 and another line l2 of degree x0 + x1 x3 x0x1x2 + x1x2 + x0x2x3 + x1x3 3. The curve C is 13-secant to l1 and 11-secant to l2.

(114(b)) Z has degree 27 and is the u-  4 3 3  x0 x0 + x2 + x3 x0 + x0x1 + x0x2 nion of a smooth irreducible 3 2  x1 x1 + x2 + x3 x0x3 + x0x1x3   4 2 2 3  curve C of degree 14 and genus  x0 + x1 x0 + x2 x0 + x0x1 + x1x2 4 3 3 11 and one line l. The curve 2x0 + x1 x1 + x3 x1 + x0x2 + x1x3 C is 11-secant to l.

(123) Z has degree 25 and is the u-  3 2  x0 + 2x1 x1x2 + x0x3 x0 + x0x1 + x0x1x3 nion of a smooth irreducible 2 3 2  x1 x1 + x1x3 x1 + x0x2   2  curve C of degree 14 and genus  x0 + x1 x0x2 + x1x3 x0x1x2 + x0x3  2 2 2 11 and one line l. The curve 2x0 + x1 x0 + x0x1 x1x2 + x1x3 C is 11-secant to l1.

(222) Z has degree 24 and is the u- nion of an irreducible curve C   of degree 12 and genus 11 and x0x2 + x0x3 x0x2 x0x1 + x1x2 + x0x3  x0x1 + x1x2 x0x1 + x0x3 x0x2 + x1x2 + x0x3  two secant lines l1 and l2 of    x0x1 + x0x3 x1x2 + x0x3 x0x1 + 2x0x2 + x0x3 degree 6 each. The curve C x0x2 + 2x1x2 x0x1 + x1x2 x0x2 + x1x2 is singular at the intersection point of l1 and l2 and is 6- secant to l1 and 6-secant to l2. Remark 4.4.1. Let us emphasize also that it would be interesting to know if, given a determinantal birational map or even a birational map with locally free sheaf of relations, its inverse is also determinantal or has locally free sheaf of relations. Our observations are that the inverse of determinantal birational map of certain type 100 CHAPTER 4. n-TO-n-TIC MAPS

(for example a quinto-quintic of type (b)) are determinantal of the same type. This leads us to the following conjecture.

Conjecture 4.4.2. The inverse of determinantal birational map of a certain type is determinantal of the same type.

4.4.2 Incursion in higher dimension

As a single example in P4, we consider the determinantal map Φ of the 4×4 minors of the matrix  2  x0 x1 x0 x0 + x1x2  x1 x2 x1 x0x1 + x1x3    x0 + x1 x3 x2 x0x2 + x1x4 .  2  x0 + 2x1 x4 x3 x0 + x0x3  2 x0 x0 x4 x1 + x0x4 Using Macaulay2, we can say that Φ has multidegree (1, 5, 8, 5, 1) (and naive multidegree (2, 7, 9, 5, 1)). However, the computation of the primary decomposition becomes unsustainable and we cannot describe yet what are the surfaces involved. We conjecture that the general determinantal map of this family has a base locus Z which is the union of an irreducible smooth surface S with resolution:

4 OP (−4) 4 4 OP (−6) ⊕ 4 4 4 4 OP (−9) ⊕ OP (−5) OP OS 0 2 4 OP (−8) ⊕ 4 OP (−7) and a plane P and that S and P intersect along a curve of degree 7 and arithmetic genus 15. ∗ n Pushing further this construction where P1 = p1P and P is (n − 2)-plane of P ∗ with n ≥ 3, and where we choose a matrix M such that p1M is general in

0 n−2 n n H IP1 (1, 1) ⊕ OP ×P (1, 1) ⊕ IP1 (2, 1) , we should be able to construct a determinantal birational map with multidegree n n n n n n−2 (1,T1 ,T2 ,...,Tn−1, 1) where Tk = k + k−1 are the coeﬃcients of the 3-Pascal triangle (see https://oeis.org/A028262). That is we should be able to construct the following multidegree

Projective space Multidegree of Φ P3 (1, 4, 4, 1) P4 (1, 5, 8, 5, 1) P5 (1, 6, 13, 13, 6, 1) P6 (1, 7, 19, 26, 19, 7, 1)

This conjecture is purely experimental and veriﬁed from P4 to P6. 4.5. HOMALOIDAL COMPLETE INTERSECTIONS 101

4.5 Homaloidal complete intersections

To ﬁnish this section let us present the following generalisation of homaloidal hypersurfaces.

Deﬁnition 4.5.1. Let n ≥ 1, 1 ≤ k ≤ n and let g1, . . . , gk ∈ k[x0, . . . , xn] be k homogeneous polynomials of degree d1, . . . , dk. Let denote   (g1)0 ... (g1)n M =  ......  (gk)0 ... (gk)n

∂gj the jacobian matrix associated to g0, . . . , gk that is (gj)i = and let φ0,..., ∂xi φ n+1 be the k × k-minors of M. We call the map ( k )

n n+1 Φ : ( k )−1 ci P P x f0(x): ... : fN (x)

n+1 where N = k − 1, the polar map of the complete intersection (g1, . . . , gk). We call the complete intersection homaloidal if Φci is birational onto its image. Remark 4.5.2. When the complete intersection is generated by n homogeneous n n polynomials g1, . . . , gn, we recover a map Φci : P 99K P . This is the reason why this notion of homaloidal complete intersection ﬁts in this chapter about locally free sheaf of relations. Indeed, since the jacobian matrix M of (g1 . . . gn) has size n × (n + 1), Hilbert-Burch theorem [Eis95, 20.15.b] states that that M is a presentation matrix for its ideal I of maximal minors if depth(I) ≥ 2. But this is just taking care of the fact that V(I) has codimension greater than 2. In other words, if V(I) has expected codimension, the sheaf of relations of I is split and a locally free resolution of I reads:

n M n+1 0 ⊕ O n (−ai) O n I(δ) 0 i=1 P P

One ﬁrst result in this context is that, when n = 2, there there is no limit in the degree of a homaloidal complete intersection.

r Proposition 4.5.3. Let r ≥ 2, the complete intersection deﬁned by x0x1, x1 + r−1 2 x0 x2 of P are homaloidal.

Proof. The polar map Φci is deﬁned by the polynomials

 r r−1 −rx1 + (r − 1)x0 x2  r−1 −x0 x1  r −x0.

It is a computation to show that Φci has an inverse deﬁned by r r−1 r r−1 −(r − 1)x2, −(r − 1)x1x2 , −rx1 + x0x2 . 102 CHAPTER 4. n-TO-n-TIC MAPS

Another result is as follows. It is classical result (see [Dés12, Subsection 4.5.2]) that there are three types of quadratic Cremona maps up to left right conjugacy, namely the standard Cremona map with representative τ = (x1x2, x0x2, x0x1), the quadratic linear system with two indeterminacy points and one base point infinitely near one of the indeterminacy points (see [Dés12, Section 1.2] for the definitions indeterminacy and infinitely near base point) with representative

0 2 2 τ = (−2x1 + x0x2 + 2x1x2, −x0x1, −x0 − 2x0x1), the quadratic linear system with three base point inﬁnitely near to each others with representative 00 2 2 τ = (x0x2 − x1, −x1x0, x0). We see from the classiﬁcation of complex homaloidal curves that only τ and τ 00 can be realised as the polar of homaloidal curve. Concerning complete intersection of 2 curves in P2, we have:

Proposition 4.5.4. All Cremona maps of P2 of algebraic degree 2 can be realised as the polar map of the complete intersection of two quadrics.

2 2 x0+x1 Proof. The quadratic map τ is the polar of the complete intersection of q1 = 2 2 2 −x0+x2 and q2 = 2 . 0 0 2 The quadratic map τ is the polar of the complete intersection of q1 = x1 +x0x2 0 and q2 = x1(x1 + x0). 00 00 2 The quadratic map τ is the polar of the complete intersection of q1 = x1 +x0x2 00 and q2 = x1x0.

n Remark 4.5.5. In general, the polar map Φci of k hypersurfaces in P factors n+1 n+1 through the Grassmanian Grassk(k ) of k-plane in k . This leads us to the following questions concerning this problem of homaloidal complete intersection: what is the multidegree of Φci with respect to the generators of the cohomology of the Grassmanian? Another perspective would also be to classify homaloidal complete intersections with ”small” singularities. Chapter 5

Free and nearly free sheaves of relations

Let X be the projective plane P2 over an algebraically closed ﬁeld k and let I be 2 an ideal sheaf generated by three global sections φ0, φ1, φ2 of OP (δ) for δ ≥ 1. In this situation we identify φ0, φ1, φ2 with homogeneous polynomials in the variables x0, x1, x2 of degree δ. Let

a (φ0 φ1 φ2) ⊕ O 2 (−i) i M 3 P O 2 I(δ) 0 i≥1 P

E be the sheafification of a minimal presentation of the ideal I = (φ0, φ1, φ2) and where E is the image sheaf of M, i.e. the sheaf of relation of I, see Definition 4.0.1. Since E is reflexive of rank 2, we have the following result. Proposition 5.0.1. [Har80] The sheaf E is locally free of rank 2.

Hence, in the perspective of studying the projectivization X of I, this is a case of interest because we can attempt to relate geometric properties of E, for instance its Chern classes, to geometric properties of X exactly in the same spirit as in Chapter4. Let us explain also our initial motivation for studying this problem. Let f ∈ k[x0, x1, x2] be a square free polynomial of degree d and let I be the jacobian ideal ∂f sheaf of f, i.e. the ideal sheaf generated by the partial derivatives fi = of f. ∂xi Deﬁnition 5.0.2. [Dim15, Deﬁnition 2.1] Letting E be the sheaf of relations of I,

2 2 the curve F = V(f) is free if E is split. If E is split and E'OP (−d1) ⊕ OP (−d2) then F is free of exponent (d1, d2). The curve F = V(f) is nearly free of exponents (d1, d2) with 1 ≤ d1 ≤ d2 if the jacobian ideal I of F has the following resolution

2 3 0 O 2 (−d2 − 1) O 2 (−d1) ⊕ O 2 (−d2) O 2 I(d − 1) 0 P P P P where d is the degree of F .

103 104 CHAPTER 5. FREE AND NEARLY FREE SHEAVES OF RELATIONS

See Definition 5.1.1 for the generalisation of these definition. Recall also the definition of Tjurina numbers.

Deﬁnition 5.0.3. Let Z = V(I) be the singular locus of a hypersurface F = V(f) and let z ∈ Z. Via a change of coordinates, suppose that z = (1 : 0 : 0). The local Tjurina number at z, denoted by τf (Z, z) is deﬁned by

∂f[ τf (Z, z) = length Ok2,z/(f[, (f[)1, (f[)2) where (f[)i = . ∂xi P The global Tjurina number of F , denoted by τf (Z) is the sum τf (Z, z) over all z ∈ Z. A result of A.A. du Plessis and C.T.C.Wall in [dPW99] identiﬁes in particular complex curves F of a given degree d with maximal possible global Tjurina number which are not cones (or else E has a non trivial factor). These are the free curves of exponents (1, d − 2). But the data of the global Tjurina number of F is equivalent to the data of the second Chern class of E via the relation 2 c2(E) = (d − 1) − τf (Z) and let us explain why. Let

3 (f0 f1 f2) 0 E O 2 I(d − 1) 0 (P6) P ∂f be the deﬁning exact sequence of E where fi = for i ∈ {0, 1, 2}. Now, consider ∂xi 2 a general morphism O 2 → I(d − 1), let W be the support of its cokernel. This P situation is summed up by the following commutative diagram:

2 2 O 2 = O 2 P P (φ φ φ ) 3 0 1 2 0 E O 2 I(d − 1) 0 P

2 OP OW 0

Since the global Tjurina number is just the length of the scheme V(I), we have that 2 2 length(W ) = (d−1) −τf (Z). Moreover there is an exact sequence E → OP → OW and recall Proposition 1.2.6 that if W has codimension 2 then the class [W ] of W 2 in the Chow group CH (P2) is the second Chern class of the dual E∨ of E. But ∨ since E has rank two, c2(E) = c2(E ) and, by identifying Chern classes and integers 2 since CH (P2) ' Z, where we fix a generator to be the class of a point in P2, we 2 have the relation c2(E) = (d − 1) − τf . Hence the identification of curves with the highest global Tjurina number is equivalent to that of the curves such that c2(E) is the smallest possible. Recall that we defined also a generalised Tjurina number in Definition 2.2.21 2 2 for any ideal sheaf I of P generated by three global sections of OP (δ). This is just the length of the scheme V(I). So for us, a main motivation is to elaborate a similar criterion to split the sheaf of relations E. Roughly, one result in this chapter is as follows: 5.1. IDENTIFICATION OF FREE AND NEARLY FREE SHEAVES 105

Theorem 5.0.4. Let I be an ideal sheaf over P2 generated by three linearly inde- 2 pendant global sections φ0, φ1, φ2 of OP (δ) and let E be the sheaf of relations of I 3 deﬁned as the kernel of the map O 2 → I(δ). P

(1) Then −c1(E) ≤ c2(E)+1 and equality holds if and only if E is free of exponents 1, c2(E) . (2) In the case c1(E) ≤ −5, E is nearly free of exponents 1, c2(E) if and only if −c1(E) = c2(E).

We emphasize that Theorem 5.0.4 is precisely a generalisation of the former result in [dPW99] since we identify free sheaves of exponents 1, c2(E) with sheaves of relations with the smallest second Chern class possible. The second part of this chapter is the classification of the reduced complex plane curves with respect to the second Chern class of their sheaves of relations, 2 that is, we classify curves of degree d and c2(E) = (d − 1) − τf . To establish it, recall that the polar degree of those curves, i.e. the topological degree of their polar 2 map is equal to (d − 1) − µf . Hence we simply consider the existing classification of plane curves with given polar degree in [FM12] and we adapt it in the cases where Tjurina numbers differ from Milnor numbers.

5.1 Identiﬁcation of free and nearly free sheaves

2 For this subsection, O stands for OP . For i ∈ {1, 2}, we denote by ci(E) the ﬁrst and second Chern classes of E and we identify the Chern classes with their degree in Z.

Deﬁnition 5.1.1. A vector bundle F of rank 2 over P2 is said to be free of ∗2 exponents (d1, d2) if there exists (d1, d2) ∈ N such that F'O(−d1) ⊕ O(−d2). It is said to be nearly free of exponents (d1, d2) if it has a graded free resolution of the form:

(φ0 φ1 φ2) 2 0 O(−d2 − 1) O(−d1) ⊕ O(−d2) F 0.

Deﬁnition 5.1.2 ([DS15]). In the case where φ0 = f0, φ1 = f1, φ2 = f2 are the partial derivatives of a given squarefree polynomial f ∈ k[x0, x1, x2], the curve F = {f = 0} is called free (resp. nearly-free) if E in (P6) is free (resp. nearly-free).

As we explained in the introduction of this section, the maximality of the Tju- rina number is equivalent to the minimality of the second Chern class c2(E) of the vector bundle E associated to F . Theorem 5.0.4 is thus a generalisation of the result of du Plessis-Wall. We emphasize that in this case c1(E) is negative and c2(E) is positive.

Proof of Theorem 5.0.4. We denote by c1 and c2 respectively the ﬁrst Chern class c1(E) and the second Chern class c2(E) of E. We let c = −1 − c1 ≥ 0 and

0 2 m = min{t ∈ Z, H P , (E(t) 6= 0}. 106 CHAPTER 5. FREE AND NEARLY FREE SHEAVES OF RELATIONS

(1) Assume that c2 ≤ c. We are going to show that the only possibility is that 0 2 c2 = c and m = 1. First, m > 0 since otherwise, if 0 6= s ∈ H (P , E) we would have had E ' O ⊕ O(−1 − c) which contradicts the fact that c2 > 0. 0 2 Now, let s ∈ H P , E(m) be a non zero section. Since m is minimal, we have the following exact sequence:

0 O(−m) E IL(m − 1 − c) 0 (E7)

where L ⊂ P2 is a 0-dimensional subscheme of length l ≥ 0. It is a computation to show that l = c2 − m(c + 1 − m) ≥ 0, and since c2 ≤ c, we have c(1 − m) ≥ m(1 − m). (5.1.1)

(i) if m = 1, then l = 0, i.e. IL(m − 1 − c) = O(m − 1 − c) and the sequence (E7) splits showing that E'O(−1) ⊕ O(−c), (ii) if m ≥ 2 then m ≥ c.

Now, assume by contradiction that m ≥ 2. First, it follows from the Riemann- Roch formula that: 8 − 2c − 3c + c2 8 − 5c + c2 χE(1) = 2 ≥ . 2 2 Hence χE(1) > 0 for all c. On the other hand, since m ≥ 2, by (ii), m ≥ c 0 2 2 2 and we have H P , E(1) = H P , E(1) = 0 where the second vanishing 2 2 0 2 ∨ follows from the ﬁrst, using Serre-duality H P , E(1) ' H P , E(c − 3) . These two vanishings contradict the fact that χE(1) > 0. Summing up, if c2 ≤ c, the only possibility is c2 = c and then E'O(−1) ⊕ O(−c) which completes the proof of (1).

(2) It is a computation to show that if E is nearly free of exponents (1, c2), then c2 = c + 1 = −c1. Now, we assume that c2 = c + 1 and that c ≥ 4 and we show that E is nearly-free of exponents (1, c2). In this case, from (E7) we have the inequality c(1 − m) ≥ m(1 − m) − 1 (5.1.2) so:

(i) m ≥ 3 implies m ≥ c (because in this case (5.1.2) is equivalent to 1 2 2 0 2 c ≤ m − 1−m ) and thus H P , E(1) = H P , E(1) = 0. But the Riemann-Roch formula implies that (c−2)(c−3) χ E(1) = 2 , hence χE(1) > 0 for c ≥ 4. As above this leads to a contradiction and so this case does not occur. (ii) m = 2 implies c ≤ 3, a case excluded by the assumption c ≥ 4. 5.1. IDENTIFICATION OF FREE AND NEARLY FREE SHEAVES 107

(iii) m = 1 implies that l = 1 where l is the length of the scheme L as in the exact sequence (E7). Now, using the resolution of a point p in P2, we get the following diagram:

α 0 O(−1) E Ip(−c) 0

β O(−1 − c)2

O(−2 − c)

where the existence of β is provided by the vanishing of Ext1(O(−1 − c)2, O(−1)) (see also [MV17] for more details in this direction). Since E is locally free of rank 2, the complex (E8) provides a locally free resolution of E showing that E is nearly-free of exponent (1, −1 − c), that is, E has the resolution:

0 O(−c − 2) O(−1) ⊕ O(−c − 1)2 E 0. (E8)

As an application we recover [DHS12, Corollary 2.6] but with a different proof. Recall that I is said to be of linear type if the projectivization X of I coincide with the Proj X˜ of the Rees algebra of I (see Definition 2.2.10). With the definitions of Milnor and Tjurina numbers given in Definition 2.2.21, recall that the ideal sheaf IZ of a 0-dimensional scheme Z in a quasi-projective variety X is of linear type if and only if µ(Z) = τ(Z).

Corollary 5.1.3. Let I = (φ0, φ1, φ2) be an ideal sheaf generated by three homogeneous polynomials of degree δ without common factor. Assume that I is of linear type then the associated map Φ is birational only if δ ≤ 2.

Proof. Indeed, letting E be as in (P6), we have that c2(E) = dt(Φ). But c1 = −δ so the only possibility to have dt(Φ) = 1 is that δ ≤ 2.

Let focus on another special resolution of a vector bundle E of rank 2 over P2. r In the following, we assume that E ⊂ O 2 for a given r ∈ N in order to have 0 P h (P2, E) = 0. Recall that such a bundle can be thought as in the exact sequence (P6).

Deﬁnition 5.1.4. We say that E is almost nearly free of exponent (d1, d2) if it has the following resolution:

0 O(−d2 − 2) O(−d1) ⊕ O(−d2) ⊕ O(−d2 − 1) E 0. 108 CHAPTER 5. FREE AND NEARLY FREE SHEAVES OF RELATIONS

Example 5.1.5. We say that a curve in P2 is almost nearly free if E in the exact sequence (P6) is almost nearly free. For instance, the curve 4 3 F = V (x1 + x0x2)(x2 + x1) is almost nearly free of exponents (2, 3) (a result we ﬁnd by computing a free resolution of the jacobian ideal of F via Macaulay2). 2 Proposition 5.1.6. Let E be a rank 2 vector bundle over P with c1(E) = −1 − c and with 1 ≤ c2(E) = c + 2. If c ≥ 5 then E is almost nearly free of exponent (1, c2(E) − 1). We give the proof of such a statement to show how the deﬁnition of almost nearly free vector bundles is analogous to the one of nearly free bundles.

0 Proof. Let m = min{t ∈ Z , h (P2, E(t)) 6= 0}. Using the sequence (E7), we have the inequality: c(1 − m) + 2 ≥ m(1 − m) that is 2 c + ≤ m 1 − m From this inegality, we get: (i) if m ≥ 4 then c ≤ m (ii) if m = 3 or m = 2 then c ≤ 4 (iii) if m = 1, then l = 2 where l is the length of the scheme L in the exact sequence (E7). Since the resolution of such a scheme L in P2 is:

0 O(−3) O(−1) ⊕ O(−2) IL 0,

as in the proof of Theorem 5.0.4, one has:

0 O(−c − 3) O(−1) ⊕ O(−c − 1) ⊕ O(−c − 2) E 0.

But now if c ≥ 5 the case (iii) cannot happen. Indeed, if it would happen, one 0 0 would have h (P2, E(1)) = h (P2, E(c − 1)) = 0. The Riemann-Roch formula gives then (c − 1)(c − 4) χ(E(1)) = > 0 2 which is impossible. So E is almost nearly free of exponent (1, c + 1). Remark 5.1.7. We could want to ﬁnd almost nearly free curves of degree d ≥ 7 with exponent (1, c2(E) − 1) But as it is stated in [dPW99, Th.3.2] in the case where there is a ﬁrst syzygy of degree one, that is with the notation of their paper, when r = 1, we have the following inequality: (d − 1)(d − 2) ≤ τ ≤ (d − 1)(d − 2) + 1 where τ is the Tjurina number of the curve. In our case, we would want a curve such that τ = (d − 1)2 − d = d2 − 3d + 1 ≤ d2 − 3d − 2 = (d − 1)(d − 2) which is impossible. 5.2. CLASSIFICATION OF CURVES BY TJURINA NUMBERS 109

5.2 Classiﬁcation of curves by Tjurina numbers

In this section, we assume that the base field k has characteristic 0. In this case, we propose a classification based on [FM12, Theorem 3.3] and [FM12, Theorem 3.4] of plane curves which are not cones with respect to the second Chern class of the sheaf of relations E of the jacobian ideal sheaf of the curve. This classification is the same except when the jacobian ideal is not of linear type, i.e. when Tjurina and Milnor numbers of the curve differ. In the table, we specify by (d1, d2)-f. or (d1, d2)-n.f. if the curves are respectively free of exponent (d1, d2) or nearly free of exponent (d1, d2). All the computation of Milnor and Tjurina numbers and of resolutions of the jacobian ideals of the curves (in order to establish in which case the curves are free or nearly free) were made by a case by case analysis of the explicit equations of the curves using Macaulay2.

Theorem 5.1. A reduced plane curve with c2(E) equal to 1 is one of the following:

(1) a smooth conic,

(2) three lines in general position

(3) the union of a smooth conic with one of its tangent

(1) (2) (3)

2 2 V(x1 + x0x2) V(x0x1x2) V(x0(x1 + x0x2) (1, 1)-n.f. (1, 1)-f. (1, 1)-f.

Proof. Let C be a curve with c2(E) = 1 and let denote by E its sheaf of relation as in (P6). By Theorem 5.0.4, the algebraic degree d of C is 0, 1, 2 or 3. Then, a case-by-case study gives the classiﬁcation.

We obtain the classification of the curves with c2(E) = 2 as follows. In [FM12, Theorem 3.3] the two authors gave the complete classification of curves with polar degree 2. Hence, the curves with jacobian ideal of linear type in this classification are curves with c2(E) = 2. Since we focus on dominant polar maps, it only remains to add the eventual curves of degrees between 0 to 4 with jacobian ideal not of linear type with polar degree 1 (there is none).

Theorem 5.2. A reduced plane curve with c2(E) = 2 with dominant polar map is one of the following:

(1) three concurrent lines and a fourth line not meeting the center point,

(2) a smooth conic and a secant line, 110 CHAPTER 5. FREE AND NEARLY FREE SHEAVES OF RELATIONS

(3) a smooth conic, a tangent and a line passing through the tangency point,

(4) a smooth conic and two tangent lines,

(5) two smooth conics meeting at a single point,

(6) an irreducible cuspidal cubic,

(7) an irreducible cuspidal cubic and its tangent at the smooth ﬂex point

(8) an irreducible cuspidal cubic and its tangent at the cusp,

(1) (2) (3)

2 2 V(x0x1x2(x0 + x1)) V(x0(x0 + x1)(x1 + x0x2)) V(x0(x1 + x0x2) (1, 2)-f. (1, 2)-f. (1, 1)-f. (4) (5) (6)

2 2 2 2 3 2 V(x0x2(x1 + x0x2)) V((x1 + x0x2)(x1 + x0x2 + x0)) V(x1 + x0x2) (1, 2)-f. (1, 2)-f. (1, 2)-n.f. (7) (8)

3 2 3 2 V(x2(x1 + x0x2)) V(x0(x1 + x0x2)) (1, 2)-f. (1, 2)-f.

As in [FM12, Theorem 3.4], we also provide the classiﬁcation of plane curves such that c2(E) = 3 and with a dominant polar map. We picture in green the plane curve whose jacobian ideal is not of linear type.

Theorem 5.3. A reduced plane curve with c2(E) = 3 is one of the types in the following tabular (up to a change of coordinates). 5.2. CLASSIFICATION OF CURVES BY TJURINA NUMBERS 111

(1) (2) (3) (4)

2 2 V(x0x1x2(x0 + V(x0x1x2(x0 + x1 + V((x1 + x0x2)(x0 + V((x1 + x1)(x0 + 2x1)) x2)) x1)(x0 + 2x1)) x0x2)x0(x0 + x2)) (1, 3)-f. (2, 2)-n.f. (2, 2)-n.f. (2, 2)-n.f. (5) (6) (7) (8)

2 2 2 2 2 2 V(x0x2x1(x1 + V((x1 + x0x2)(x1 + V(x0(x1 +x0x2)(x1 + V(x0(x1 + x0x2)) 2x0x2)) 2 2 x0x2 + x0)) x0x2)(x1 + 2x0x2)) (1, 3)-f. (1, 3)-n.f. (1, 3)-f., dt = 2 (1, 3)-f. (9) (10) (11) (12)

2 2 3 3 2 3 2 V((x1 + x0x2)(x1 + V((x1 + x0)(x1 + V((x1 + x0x2)(x2 + V(x0x2(x1 + x0x2)) x0x2 + x1x2)) 2 3x1 − 2x0)) x0x2)) (2, 2)-n.f. (1, 3)-n.f. (2, 2)-n.f. (1, 3)-f. (13) (14) (15) (16)

3 2 3 2 2 2 2 V(x0x1(x1 + x0x2)) V(x1x2(x1 + x0x2)) V(x1x2 −x0(x0 +x2)) V((x0 − x1)(x1x2 − 2 x0(x0 + x2))) (1, 3)-f. (1, 3)-f. (2, 2)-n.f. (17) (18) (19) (20)

2 3 4 4 2 4 2 V(x2(x1x2 − V(8x0x1 − 3x1 + V(x1 − 2x0x1x2 − V(x1 − 2x0x1x2 − 2 2 3 3 2 2 3 2 2 x0(x0 + x2))) 12x0x1x2 − 4x1x2 + x1x2 + x0x2) x1x2 + x0x2) 2 2 4x0x2) (2, 2)-n.f. (2, 2)-n.f. singular points : singular point : (1 : 0 : 0), (0 : 0 : 1) (1 : 0 : 0), (2, 2)-n.f. (2, 2)-n.f. 112 CHAPTER 5. FREE AND NEARLY FREE SHEAVES OF RELATIONS

(21) (22) (23) (24)

4 3 4 3 4 3 4 3 3 V(x1 − x0x2) V(x2(x1 − x0x2) V(x0(x1 − x0x2)) V(x1 − x1x2 + x0x2) singular point : (1, 3)-f. (1, 3)-f. singular point : (1 : 0 : 0), (1, 3)-n.f. (1 : 0 : 0), (2, 2)-n.f. We end this chapter by giving the curves with polar degree 3 and naive polar degree c2(E) = 4.

2 2 2 2 2 2 2 V((x1 + x0x2)(x0 + V(x2(x1 +x0x2)(x1 + V((x1 + x0x2)(x1 + V(x0(x1 +x0x2)(x1 + x1)(x0 + 2x1)x0) x0x2 + x1x2)) 2 2 2 2 x0x2 + x0)(x1 + x0x2 + x0)(x1 + 2 2 x0x2 + 2x0)) x0x2 + 2x0)) (2, 2)-f. (2, 2)-f. (1, 4)-f. (1, 5)-f.

3 4 2 4 2 V(x0(x0 + x1)(x1 + V((x0 − x1)(x0 + V(x2(x1 − 2x0x1x2 − V(x2(x1 − 2x0x1x2 − 2 2 2 3 2 2 3 2 2 x0x2)) x1)(x1x2 − x0(x0 + x1x2 + x0x2)) x1x2 + x0x2)) x2))) (2, 2)-f. (2, 2)-f. (2, 2)-f. (2, 2)-f.

4 3 V(x2(x1 − x1x2 + 3 x0x2)) (2, 2)-f. Chapter 6

Zero-dimensional base locus

n In this section, we focus on the naive multidegree of a rational map Φ : X 99K P with a zero-dimensional base locus Z in a smooth quasi-projective variety X of dimension n. Let us first present the problem we deal with. As we explained in n n n Chapter4 in the case X = P , if the sections φ0, . . . , φn defining Φ : P 99K P are the n + 1 minors of a matrix M of size (n + 1) × n, it is particularly easy to compute its naive multidegree. Actually this can be done in two ways. The first one is to consider a presentation of the base ideal sheaf IZ

n M n n+1 0 ⊕ O (−ai) O n IZ (δ) 0. i=1 P P

The nth naive projective degree of Φ is then the length of the intersection of n n P general generators λiφi where λi ∈ k for any i ∈ {0, . . . , n} after removing the i=0 points already in Z (see Section 1.3). Hence the nth projective degree of Φ is the length of the scheme V(s) deﬁned in the following exact sequence:

n O n I (δ) O (s) P Z V 0

n where the morphism O n → I (δ) is general. Now consider this morphism in the P Z following commutative diagram:

n n O n = O n P P

n M n n+1 0 ⊕ O (−ai) O n IZ (δ) 0 i=1 P P

n OP Os 0

0 0

113 114 CHAPTER 6. ZERO-DIMENSIONAL BASE LOCUS

n So V(s) is also the support of the cokernel of the map E = ⊕ O n (−ai) → O n . i=1 P P In other words, the class [V(s)] of V(s) in the Chow ring of Pn is the nth Chern class of the dual E∨ of E, i.e. s is a cosection of E. Since E is in particular locally free, we have that s is the nth Chern class of E. As such, its length (as a 0-cycle) n Q is particularly easy to compute in this case, it is ai. i=0 The second way to compute this projective degree is to consider the projec- n n tivization X of IZ in P × P . Since it is a complete intersection of cohomological n Q n th class (aih1 + h2) in CH(P × P ), we recover that the n naive projective degree i=1 n Q is ai. i=0 The main goal of this chapter is to establish if those two ways of computing the naive projective degrees, namely considering cosections of E or the projectivization X, always coincide even if E is not locally free. We summarize our result as follow, see Theorem 6.1.1 for a more precise statement. Theorem 6.0.1. In the case when the base locus Z is 0-dimensional, the length of n+1 the zero scheme of a cosection of the kernel E of the evaluation map O n → I(δ) P is equal to the nth naive projective degree. In the case when the base locus Z is 0-dimensional, we emphasize that the possible torsion components of X only interfere with the nth projective degree, that is, projective degrees and naive projective degrees coincide up to the nth projective degree. That is why we only consider two naive topological degrees, see Deﬁnition 6.1.4 and Deﬁnition 6.1.5 which have to be understood as the two ways of computing the nth projective degree of Φ.

6.1 First and second naive degrees

We let X be an n-dimensional smooth quasi-projective variety over k and we let n Φ: X 99K P be a rational map with zero-dimensional base locus Z determined by a n + 1-dimensional subspace V of global sections of a line bundle L over X. Our aim is to read off the topological degree dn(Φ) of Φ from properties of the ideal sheaf I of Z, more precisely from the sheaf of relations E defined as the kernel of the canonical evaluation map ev : OX ⊗ V → I ⊗ L (Definition 4.0.1). We set also the notation as in the following diagram

n PX

p1 p2 (D1)

π1 X˜ π2

σ1 σ2 Φ X Pn 6.1. FIRST AND SECOND NAIVE DEGREES 115

˜ where X is the projectivization of IZ and X is the blow-up of X along Z (recall Proposition 2.1.8 that X˜ is isomorphic to the graph Γ of Φ). By construction the topological degree of Φ is equal to that of the restriction to X˜ of the lift n π2 : X → P(V) of Φ. In other word dt(Φ) = deg c1(OX(1) X˜ ) . In this context, we can also consider two other related notions of ”naive” topo- n logical degrees: the degree deg c1(OX(1)) of π2, which is the deﬁnition of the nth naive projective degree of Φ and the algebraic degree of Φ minus the length of Z. In Proposition 6.2.2 (i) we show that the second one coincides with the degree c of the 0-cycle [V s(E) ] associated to the scheme of zeros of a general cosection c s(E): E → OX of E. Our main result, proven in Subsection 6.1.3, asserts in particular that these two naive topological degrees coincide. It also elucidates the relation between these degrees and the topological degree of Φ:

Theorem 6.1.1. With the notation above, X is equidimensional of dimension n c n and [V s(E) ] = π1∗c1 OX(1) . As a consequence c n dn(Φ) = deg [ ( s(E))] − deg c1(O (1) ) . V X TZ where TZ is the torsion part of X, see Notation 2.2.20. Recall that the two classical invariants of singularities of a hypersurface F = {f = 0} is the global Tjurina number τf (Z) of F and the global Milnor number µf (Z) (see Definition 16 and Definition8). Actually, both Milnor and Tjurina numbers depend on the scheme structure of the singular locus, and, in this sense, they can be defined also for zero-dimensional subschemes Z unrelated to singular hypersurfaces. Having this in mind, recall the result Theorem 10

Theorem 6.1.2. Let f ∈ C[x0, . . . , xn] be a square free homogeneous polynomial n of degree d and let Φf be the polar map of F = V(f) ⊂ P . Assuming that F has ﬁnite base locus, we have:

n dn(Φf ) = (d − 1) − µf (Z) (6.1.1) In comparison to (6.1.1), we can formulate the following result. Corollary 6.1.3. Formula (6.1.1) holds for any 0-dimensional subscheme Z de- ﬁned by n + 1 global sections of a line bundle L over a smooth quasi-projective n-variety X and over any algebraically closed ﬁeld. This corollary follows from the observation that Tjurina numbers compute the n n degree of c1 OX(1) whereas Milnor numbers compute the degree of c1 OX˜ (1) . As an immediate application we recover the identity (0.0.1) from the equalities

c n n deg [ ( s(E))] = (d − 1) − τ(Z) and deg c1(O (1) ) = µ(Z) − τ(Z) V X TZ where τ(Z) and µ(Z) are the generalised Tjurina and Milnor numbers. In our setting, recall that X is equidimensional of dimension n (see Propo- n sition 2.2.19) so c (O n (1) ) is a 0-cycle on and we can set the following 1 PX X deﬁnition. X

n Deﬁnition 6.1.4. With the notation in (D1) the degree of c (O n (1) ) is called 1 PX the ﬁrst naive topological degree of Φ. X 116 CHAPTER 6. ZERO-DIMENSIONAL BASE LOCUS

Intuitively, the difference between the first naive topological degree and the actual topological degree reflects a difference between the symmetric algebra and the Rees algebra, see Proposition 6.2.2 below for a precise statement. n+1 Now, let E be the kernel of the evaluation map ev : OX → I ⊗ L and let n+1 c α : OX → OX be a general map. Since E has rank n, the zero locus V( sα) of c the composition sα = α ◦ γ is a 0-dimensional subscheme of X.

γ n+1 ev 0 E OX I ⊗ L 0 c α sα OX

In the proof of Theorem 6.1.1, we will establish in particular that the cycle class c c [V( sα)] of V( sα) is independent on the choice of a general map α so, anticipating, we set the following deﬁnition.

Deﬁnition 6.1.5. The second naive topological degree of Φ is the degree of the c c 0-cycle [V s(E) ] of a general cosection s(E) of E.

c Remark 6.1.6. If E is locally free, [V s(E) ] simply coincides with the top Chern ∨ ∨ class cn(E ) of E . This is no longer true when E is not locally free. For instance 2 2 the sheaf E of relations of the ideal sheaf I = (x1 − x1x3, x2 − x2x3, x1x2, x0x3) 3 ∨ c of P satisﬁes c3(E ) = 4 whereas deg [V( s(E))] = 2 as we can check from the resolution of E:

2 2 3 3 3 3 0 OP (−3) OP (−1) ⊕ OP (−2) E 0.

6.1.1 Importance of the subregularity of the symmetric algebra Recall the settings of Theorem 6.1.1, we assume that n ≥ 2, codim(Z) = n and that the map Φ is dominant. By deﬁnition, the ﬁrst naive topological degree is the length of the 0-scheme n W of a general section of OX(1) . Our strategy to show Theorem 6.1.1 is now to push forward the following exact sequence:

0 K On O (1) O (1) 0 (E9) X X W

n where K is by deﬁnition the kernel of the map O → OX(1). So, applying π1∗ to 1 1 X (E9) and assuming that R π1∗ K = R π1∗ IW (1) = 0, we have

On X π1∗OX(1) π1∗OW (1) 0.

We emphasize that I is not locally free so π1∗(OX(1)) might a priori be diﬀerent from I (see Stack project, 26.21. Projective bundles, example 26.21.2). However our strategy is to prove that these coincide in this case. 6.1. FIRST AND SECOND NAIVE DEGREES 117

ι n We use the same notation for the sheaves and their push forward by X ,→ PX . 1 1 Thus, the strategy is to ensure that R p1∗ K = R p1∗ IW (1) = 0 and that p1∗ OX(1) = I ⊗ L in order to get the sequence:

n OX I ⊗ L p1∗OW (1) 0. (E10)

c As we will explain below, [p1∗OW (1)] will turn out to be precisely the cycle [V( sα)] which by deﬁnition veriﬁes the following exact sequence:

c sα O c E OX V( sα) 0.

This will show eventually Theorem 6.1.1.

6.1.2 Cohomological preliminaries Lemma 6.1.7. The following vanishings hold:

1 (i) R p1∗IX(1) = 0, i+1 (ii) R p1∗OX(−i) = 0 for every i ∈ {0, . . . , n − 1}, i (iii) R p1∗OX(−i) = 0 for every i ∈ {1, . . . , n − 1}.

Proof. Under the assumption that dim(Z) = 0, by Corollary 3.2.17, the ideal IX has a locally free resolution of the following form:

0 Gn+1 Gn ... G2 G1 IX 0 (G•)

i ∗ ∗ 0 where Gi = ⊕ p Gij ⊗O n (−j) when i ∈ {1, . . . , n} and Gn+1 = p Gn+1 ⊗O n (−1) j=1 PX PX 0 for some locally free sheaves Gij and Gn+1 over X. 1 Now, a diagram chasing in (G•) shows that R p1∗IX(1) = 0 provided that k R p1∗ Gk(1) = 0 for all k ∈ {1, . . . , n + 1}. By Proposition 3.1.6, those vanishings are veriﬁed since:

k n n • H P , OP (−j + 1) = 0 for all k ∈ {1, . . . , n} and all j ∈ {1, . . . , k},

n+1 n n • H P , OP (−2) = 0 The only non trivial case to check is when k = n. But:

n n 0 n ∨ n n H P , OP (−j + 1) ' H P , OP (j − n − 2) = 0 because j ≤ n. For (ii) and (iii), since O = O n /I , the assertions follow from the same X PX X argument after twisting the complex (G ) by O n (−i) for every i ∈ {0, . . . , n − • PX 1}. Lemma 6.1.8. We have p1∗ OX(1) = I ⊗ L. 118 CHAPTER 6. ZERO-DIMENSIONAL BASE LOCUS

Proof. First, O n (1) being the relative ample line bundle of the projective bundle PX n n+1 n+1 = O , we have p O n (1) = O . PX P X 1∗ PX X ∗ Moreover, since I (1) is the image of the canonical map p E → O n (1), we let X 1 PX H be the kernel of this surjection and we write the exact sequence:

∗ 0 H p1E IX(1) 0.

∗ 1 ∗ Since p1∗p1E'E and R p1∗p1E = 0, applying p1∗ to this exact sequence, we get: 1 0 p1∗H E p1∗IX(1) R p1∗H 0. (a)

1 Also, since we proved that R p1∗IX(1) = 0, applying p1∗ to the canonical exact sequence

0 I (1) O n (1) O (1) 0 X PX X we get n+1 0 p1∗IX(1) OX p1∗OX(1) 0. (b)

The exact sequences (a) and (b) ﬁt into the following commutative diagram:

p1∗H 0

E ' E

n+1 0 p∗IX(1) OX p1∗OX(1) 0 = 1 0 R p1∗H IZ ⊗ L p1∗OX(1) 0

0 0

where (a) is the left column, (b) is the central row and the map IZ → p1∗OX(1) in the bottom row is the canonical morphism associated to the projectivization of IZ .

This morphism is an isomorphism over X\Z and therefore IZ ⊗ L → p1∗OX(1) is 1 injective because IZ is torsion free. Hence p1∗H' 0 ' R p1∗H and p1∗OX(1) ' IZ ⊗ L.

6.1.3 Proof of Theorem 6.1.1

As above, let W ⊂ X be the intersection of X with n general relative hyperplanes n n of PX so that [W ] = c1 OX(1) . 6.1. FIRST AND SECOND NAIVE DEGREES 119

Proof of Theorem 6.1.1. Consider the following exact sequence:

β0 0 K On O (1) O (1) 0 X X W

IW (1) (Kz)

0 0

We claim that 1 1 R p1∗ IW (1) = R p1∗ K = 0. To prove it, ﬁrst observe that the Koszul complex

n 0 O (−n + 1) ... O (−1)(2) On I (1) 0 X X X W is exact. Indeed X has dimension n since, by Lemma 3.2.8, X decomposes as the ˜ union of X, the blow-up of X at I, and the torsion part TZ , possibly empty, whose n 0 n reduced structure is PZ0 for a set Z ⊂ Z. Moreover the intersection W in PX of and n generic divisors in |O n (ξ)| (since O n (ξ) is very ample along the ﬁbres) X PX PX has codimension n in X. Hence the Koszul complex of these sections, restricted to X is exact. Then, cutting the Koszul complex into short exact sequence and taking direct images, we see that the desired vanishing holds if we show:

i+1 • R p1∗OX(−i) = 0 for every i ∈ {0, . . . , n − 1}, i • R p1∗OX(−i) = 0 for every i ∈ {1, . . . , n − 1}. On the other hand, this last vanishing is precisely the content of Lemma 6.1.7 (ii) and (iii). Since p On 'On , p O (1) ' I ⊗ L,R1p I (1) = 0 and R1p (K) = 0, 1∗ X X 1∗ X 1∗ W 1∗ pushing forward by p1 the exact sequence (Kz), we obtain the following commutative diagram:

0 p1∗K

n n OX = OX 0 β p1∗β n+1 (φ0 ... φn) 0 E OX I ⊗ L 0 (D2)

= α c sα E OX p1∗OW (1) 0

0 0 where the map α is deﬁned by the diagram (D2) as the cokernel of map β induced 0 n+1 α from p1∗(β ). The composition E → OX → OX gives rise to a cosection sα and, by the bottom row of (D2), we have p1∗(OW ) 'OV (sα). Therefore,

[V (sα)] = p1∗[W ]. (6.1.2) 120 CHAPTER 6. ZERO-DIMENSIONAL BASE LOCUS

n Now, by deﬁnition we have dn(Φ) = deg(ξ X˜ ). Also, we have

n n deg(W ) = deg(ξ ˜ ) + deg(ξ ) X TZ hence d (Φ) = deg(W ) − deg(ξn ). n TZ So the statement of the theorem amounts to deg(W ) = deg V(sα) . This is guaranteed by (6.1.2). Since all the general map α as in (E10) can be obtained as cokernel of a general n n+1 c map β : OX → OX ,[V( sα)] does not depend on the general map α so that we c c can write [V s(E) ] for a general cosection s(E). The fact that deg(W ) = deg(p1∗W ) comes from the decomposition of W . ˜ Indeed, X decomposes into the graph X and the torsion part TZ supported on n . Hence, we have the equality PFittn−1(Z) n n n [W ] = [ ] · c O n (1) = [X˜] · c O n (1) + [ ] · c O n (1) . X 1 PX 1 PX TZ 1 PX

Since X˜ is irreducible and σ1 : X˜ → X birational, we have n n deg [X˜] · c O n (1) = deg σ ([X˜] · c O n (1) ) . 1 PX 1∗ 1 PX

Moreover, as a consequence of Theorem 6.1.1 we have:

c n d (Φ) = deg [ ( s(E)] − deg p ([ ] · c O n (1) ) . t V 1∗ TZ 1 PX

6.2 Study of homaloidal hypersurfaces

6.2.1 Measure of the diﬀerence between Rees and symmetric algebras We relate now the topological degree and the naive topological degree with the notions of Milnor and Tjurina numbers. For the rest of this section, I is the ideal 0 of a rational map Φ = (φ0 : ... : φn) associated to an n+1-subspace V of H (X, L) where L is a line bundle over X. We denote by Z the base scheme V(I) in X and we assume that dim(Z) = 0.

Generalized Milnor and Tjurina numbers

n n Notation. We set temporarily as a notation that δ = deg c1(L) which as to n be understood as δ = deg c1(L) when X is the projective space P . Deﬁnition 6.2.1. With notation as in Notation 2.2.20, for every z ∈ Z, put:

• τ(Z, z) = length(OZ,z)

• µ(Z, z) = τ(Z, z) + deg(Tz). 6.2. STUDY OF HOMALOIDAL HYPERSURFACES 121

We let τ(Z) = P τ(Z, z) and µ(Z) = P µ(Z, z). z∈Z z∈Z As a direct application of Theorem 6.1.1, we obtain:

Proposition 6.2.2. The following equalities hold:

c n (i) deg [V( s(E))] = δ − τ(Z) n (ii) dt(Φ) = δ − µ(Z)

c (iii) dt(Φ) − deg( s(E)) = µ(Z) − τ(Z) = deg(T ) = deg(p1∗T ).

Proof. Looking back at the diagram (D2), we see that V(sα) has the following presentation:

n n P P sα = ( ai1φi ... ainφi) n i=0 i=0 O OX I ⊗ L V(sα) 0 where (aij)0≤i≤n,1≤j≤n is an (n + 1) × n general matrix with entries in the field c c k. Since by definition, [V( s(E))] = [V(sα)] we have that deg [V( s(E))] = length(O ) = δn − τ(Z) by definition of τ(Z). V(sα) The equalities (ii) and (iii) follow in the same way from the definition of µ(Z) ˜ and τ(Z) and from the decomposition of X as the union of X and TZ . The number µ(Z) being the sum of the numbers µ(Z, z), we explain how to computationally compute the number µ(Z, z).

Proposition 6.2.3. Let (aij)0≤i≤n,1≤j≤n be an (n + 1) × n general matrix with n n P P entries in the ﬁeld k. Then, denoting by ( ai1φi,..., ainφi)z the localisation i=0 i=0 at z, we have:

n n X X µ(Z, z) = length(OX,z/( ai1φi,..., ainφi)z). i=0 i=0 Proof. First, since the formation of the symmetric algebra commutes with base change, we can suppose that Z consists of a single point z so that µ(Z) = µ(Z, z). Now recall that dt(Φ) can be computed in the following way. A general point n n P y ∈ P is the intersection of n general hyperplanes Lj : aijxi = 0, that is, the i=0 data of an (n + 1) × n general matrix N with entry in the ﬁeld k. The preimage n n 0 P P of y by Φ is contained in the scheme F = V( a1jφj,..., anjφj). Hence, to j=0 j=0 compute the topological degree of Φ, it remains to remove the points of F0 in the base locus. So

n n n X X dt(Φ) = length(F) = δ − length(OX,z/( ai1φi,..., ainφi)z), i=0 i=0 122 CHAPTER 6. ZERO-DIMENSIONAL BASE LOCUS

n n and since dt(Φ) is also equal to δ − µ(Z) = δ − µ(Z, z), we have that

n n X X length(OX,z/( ai1φi,..., ainφi)z) = µ(Z, z). i=0 i=0

As explained to us by Laurent Bus´eafter the defense of this thesis, µZ,z can also be interpreted as the multiplicity of the Hilbert-Samuel polynomial of the localization of I at z (see [Eis95, Chapter 12] for the related deﬁnitions). Remark 6.2.4. As it is explained in Example 1.3.9, in practice via Macaulay2, letting n n 0 X X F = V( a1jφj,..., anjφj) j=0 j=0 as in the proof of 6.2.3, the preimage of y is equal to the scheme

n n X X ∞ F = V([( a1jφj,..., anjφj):(φ0, . . . , φn)] ) j=0 j=0 where, given two ideals J and J 0 of a ring R, we let [J : J 0∞] is the saturation of J by J 0.

The polar case

In the polar case, X is the projective space Pn over k. Deﬁnition 6.2.5. Let F = {f = 0} be a hypersurface in Pn where f is a homoge- ∂f neous polynomial of degree d in k[x0, ··· , xn]. Let fi = and I = (f0, . . . , fn) be ∂xi n the ideal sheaf in OP generated by the partial derivatives of f, called the jacobian ideal of f. Recall that we call the map Φf associated to I the polar map. The topological degree of Φf is called the polar degree of F .

In order to use the Euler identity d · f = x0f0 + ... + xnfn, we suppose in the sequel that the characteristic of the base field does not divide the degree of the polynomial f defining the hypersurface F . We also always assume that the jacobian ideal I of F is zero-dimensional. We recall the classical definition of Milnor and Tjurina numbers.

Deﬁnition 6.2.6. Let z ∈ Z = V(I) and via a change of coordinates, suppose that z = (1 : 0 : ... : 0). Set g[ ∈ k[x1, . . . , xn], the usual deshomogeneisation of a homogeneous polynomial g ∈ k[x0, ··· , xn] in the chart {x0 6= 0}. The local Tjurina number at z, denoted by τf (Z, z) is deﬁned as

∂f[ τf (Z, z) = length Okn,z/(f[, (f[)1,..., (f[)n) where (f[)i = . ∂xi

The local Milnor number at z, denoted by µf (Z, z), is deﬁned as

∂f[ µf (Z, z) = length Okn,z/((f[)1,..., (f[)n) where (f[)i = . ∂xi 6.2. STUDY OF HOMALOIDAL HYPERSURFACES 123

The global Tjurina number of F , denoted by τf (Z) (resp. global Milnor number P P of F , denoted by µf (Z)) is the sum τf (Z, z) (resp. µf (Z, z)) over all z ∈ Z.

We explain now how the numbers µ(Z) and τ(Z) defined in Definition 6.2.1 coincide with the usual definitions of Milnor and Tjurina number given in Defini- tion 6.2.6.

Proposition 6.2.7. Let F = {f = 0} be a reduced hypersurface in Pn where f is a homogeneous polynomial in k[x0, ··· , xn] of degree d. Let z ∈ Z = V(I) then:

τ(Z, z) = τf (Z, z) and µ(Z, z) = µf (Z, z).

Proof. Via a change of coordinates, we can suppose z = (1 : 0 : ... : 0). The deshomogenisation of the Euler identity in the chart {x0 6= 0} is:

n X d · f[ = (f0)[ + xi(fi)[ i=1 and (fi)[ = (f[)i for 1 ≤ i ≤ n. The equality

((f0)[,..., (fn)[) = (f[, (f[)1,..., (f[)n) implies that τ(Z, z) = τf (Z, z). For the Milnor number, we let A = (aij)0≤i≤n,1≤j≤n a general (n+1)×n matrix with entries in the ﬁeld k. By Proposition 6.2.3,

n n X X n µ(Z, z) = length OP ,z/( ai1fi,..., ainfi)z . i=0 i=0

By localisation at z, we have that µ(Z, z) = length(OMA ) where MA is deﬁned as the cokernel of the following composition map:

(f . . . f ) n A n+1 0 n z Oz Oz Oz OMA 0 whereas µf (Z, z) = length(OM ) where M is deﬁned as the cokernel of the following composition map:

  0 0    1       0     0    0 0 1 (f . . . f ) n n+1 0 n z Oz Oz Oz OM 0.

But, since rank(A) = n, we have length(OMA ) = length(OM ).

In the case when τ(Z, z) = µ(Z, z) for a point z ∈ Z, Z is also called quasi- homogeneous at z in [Sai80]. As an application of the previous proposition, we recover a result originally proved over the ﬁeld C in [DP03]. 124 CHAPTER 6. ZERO-DIMENSIONAL BASE LOCUS

Proposition 6.2.8. Let F = {f = 0} ⊂ Pn be a reduced hypersurface of degree d over an algebraically closed ﬁeld k. Let Φf = (f0 : ... : fn) be the polar map of f and assume that V(f0, . . . , fn) is ﬁnite. Then n dt(Φf ) = (d − 1) − µf (Z).

Proof. Since f has degree d and V(f0, . . . , fn) is ﬁnite, Proposition 6.2.8 follows from Proposition 6.2.7 and Proposition 6.2.2 (ii) since the polynomials fi have degree d − 1.

6.2.2 Proofs of Proposition 19 and Proposition 20 In characteristic 3, a homaloidal curve of degree 5 We give eventually the proof of Proposition 19. Recall the result.

2 2 2 Proposition 6.2.9. The curve F = V (x1+x0x2)x0(x1+x0x2+x0) is homaloidal if and only if the base ﬁeld k has characteristic 3, in which case the inverse of the polar map is

2 2 3 4 3 2 3 4 2 3 Ψ = (x1x2 + x0x2 + x2 : −x1x2 − x0x1x2 − x1x2 : −x1 − x0x1x2 + x0x2)

Proof. Here all the computations of resolutions, descriptions of the projectivization of the ideal sheaf, of the graph of the polar, of the torsion components or of the Milnor/Tjurina numbers were made using Macaulay2. The curve F is defined over Z hence over Fp for every p. Given the characteristic p of k, we will denote the polar of F by Φp. First, let us explain why the polar of F cannot be birational if p > 20. The idea here is that the polar has the same behaviour in high enough characteristic than in characteristic 0. In order to do so we follow the presentation in [Ngu16]. When p > 20, remark that the base locus of Φp is support over z = (0 : 0 : 1) because the reduction of the equation of F modulo p does not affect its coefficients. Now, recall that the conductor invariant δ(Z, z) is defined as the length of the quotient module OF,z/OF,z, where OF,z is the normalization of the local ring OF,z. Let r(F, z) denote the number of local branches of F at z. Using étalecohomology, Deligne showed (cf. [Del73], [MHW01]) that

µ(Z, z) = 2δ(Z, z) − r(F, z) + 1 + Sw(F, z) where Sw(F, z) denotes the Swan conductor of F at z (see [Del73, 1.7, 1.8] for the deﬁnition). In our case, we have after computation that δ(Z, z) = 8 and r(F, z) = 3 and by [Ngu16, Corollary 3.2], we have that Sw(F ) = 0 if p > κ(F ) where κ(F ) = 2 dim OP ,z)/(f[, αfx0 + βfx1 ) where fx0 and fx1 are the partial derivatives of the 2 2 2 deshomogeneisation f[ of (x1 + x0x2)x0(x1 + x0x2 + x0) with respect of x2 and (α : β) ∈ P1 is generic. Hence κ(F ) ≤ 20. So, if p > 20, we have that µ(Z, z) = 14 so d2(Φp) = 2 and F is not homaloidal. Concerning the remaining case 2 ≤ p ≤ 20, we proceed as follows. The resolution of the jacobian ideal I over Q is as follows: 6.2. STUDY OF HOMALOIDAL HYPERSURFACES 125

 3 2 2  0 2x0+4x0x1+4x0x2    x −x3   0 1  2 2 2 2 −2x1 −6x0x −8x x2−8x x2−6x0x 0 O(−1) ⊕ O(−3) 1 0 1 2 O3 I(4) 0 (R11)

2 where we denote O for the sheaf OP . By a case by case computation, we observe that for every prime p 6= 2 and p 6= 5 the reduction modulo p of (R11) provides a resolution of Ip = I ⊗Z Fp. In every characteristic 20 ≥ p ≥ 3 diﬀerent from 5, Fitt1 Ip = (x0, x1) so Ip is not a complete intersection and P(Ip) has a torsion component above the point z = (0 : 0 : 1) ∈ P2. Moreover, in characteristic other than 2 and 5, the resolution of Xp = P(Ip) embedded in Pn × Pn is as follow:

0 O(−4, −2) O(−1, −1) ⊕ O(−3, −1) IXp 0

n n where O stands for OP ×P and we wrote to the right the shift in the variables of the second factor of the product Pn × Pn. From this resolution, we can compute that τ(Z, z) = 13 in every characteristic other that 2 and 5. In characteristic 3, I3 has the following resolution:

 3 2 2  0 x0 − x0x1 − x0x2 3 x0 x1  2 2 x1 −x0x2 − x1x2 3 0 O(−1) ⊕ O(−3) O I3(4) 0.

The diﬀerence in characteristic 3 comes from the multiplicity of the torsion component in X3. Indeed, the torsion component TZp has the following resolution over Fp for 5 < p ≤ 20.

2 I 0 O(−2, 0) O(−1, 0) TZp 0 whereas in characteristic 3, it has resolution:

O(−3, 0)2 O(−2, 0)3 I 0 O(−3, −1) ⊕ ⊕ TZ3 0. O(−2, −1)2 O(−1, −1)

Given the generators of I (respectively I in characteristic 5 < p ≤ 20), we TZ3 TZp compute that the multidegree of TZ3 (resp. TZp ) is (2, 0, 0) (resp. (1, 0, 0)). Hence the diﬀerence µ(Z)−τ(Z) is equal to 2 in characteristic 3 (resp. µ(Z)−τ(Z) = 1 if p > 5) so the Milnor number µ(Z, z) is equal to 15 and d2(Φ3) = 1 in characteristic 3 or else µ(Z, z) = 14 and d2(Φp) = 2 for p > 5. In characteristic 3, the polar map can be written

4 3 2 3 3 2 4 2 2 3 Φf = (x1 + x0x2 + x0x1x2 : −x0x1 + x0x1 + x0x1x2, x0 − x0x1 − x0x2). 126 CHAPTER 6. ZERO-DIMENSIONAL BASE LOCUS

and it is a computation to check that Ψ is the inverse of Φf . As we will see in Remark 7.2.6, the inverse Ψ of Φ is actually the polar map of the dual curve F . When p = 2 or 5, the polar of F is not dominant.

Remark 6.2.10. What we did is to deepen the multiplicity of the torsion component by specializing the resolution of I over Z modulo a prime p for which some monomials of the presentation matrix disappear (here p = 3 works). More precise- 2 t ly, let Ot be the subscheme of P deﬁned by (x0, x1) . Hence M O1 = 0 in any characteristic. But if char(k) 6= 3, M O2 has two non zero columns, one of linear entries and one of quadratic entries whereas if char(k) = 3 the quadratic column- s vanishes. Hence (I (4) ) is diﬀerent with the characteristic of k. In other P Z O2 words, in characteristic 3, the torsion part TZ is not equal scheme-theoretically to n contrary to what happens in greater characteristic. It is not clear if such PFittn−1 I an example is sporadic or not which leads to the following questions.

Problem F. Given a ﬁeld k of positive characteristic, are the homaloidal curves 2 of Pk of bounded degree? What is the classiﬁcation of such curves?

Reduction problem in positive characteristic Proposition 6.2.11. Let k be an algebraically closed ﬁeld of characteristic 101.

3 2 50 3 2 51 (i) The curve V z(y + x z) has polar degree 2 whereas V z (y + x z) has polar degree 1.

3 2 2 3 (ii) The curve V (y +x z)(y +xz) has polar degree 5 whereas the curve V (y + x2z)31(y2 + xz)4 has polar degree 3.

The analysis of the presentation of the jacobian ideal gives also an easy way to construct examples of non reduced plane curves in positive characteristic where the topological degree is not preserved by reduction. It suffices to compute the presentation matrix of the jacobian ideal and adjust the characteristic of the field in order to modify the first syzygy matrix.

Proof. All the computations of resolution were made using Macaulay2. Both curves are deﬁned over Z and as in the proof of Proposition 6.2.9, the idea is to take reduction modulo the prime p = 101 of the resolution of their jacobian ideal over Z to get a resolution over Fp. We give the complete argument for Item (i). Item (ii) is similar and left to the reader. As in the proof of Proposition 6.2.9, Ip stands for I ⊗Z Fp. 3 2 The jacobian ideal of V z(y + x z) = 0 has resolution

Φred 3 0 O(−1) ⊕ O(−2) O I101(3) 0,

IX101 has the following resolution:

O(−1, −1)

0 O(−3, −2) ⊕ IX101 0. O(−2, −1) 6.2. STUDY OF HOMALOIDAL HYPERSURFACES 127

There is no torsion component above the point z = (1 : 0 : 0) and so the corresponding polar map has topological degree 2. 50 3 2 51 But the jacobian ideal of the curve V z (y + x z) has resolution

3 Φ 0 O(−1) ⊕ O(−2) O I101(3) 0

and IX101 has the following resolution:

O(−1, −1)

0 O(−3, −2) ⊕ IX101 0. O(−2, −1)

There is a torsion component above the point z = (1 : 0 : 0), what we can see from the resolution of X˜:

O(−1, −1) ⊕ 2 0 O(−2, −2) O(−2, −1) IX˜ 0. ⊕ O(−1, −2)

The polar map of the latter curve is given by

(x : y : z) 7→ (xz2 : −49y2z : 50y3) and its inverse is (x : y : z) 7→ (−37xz2 : −3y2z : y3).

Chapter 7

Inverse of a birational map

n n Suppose we are given a birational map Φ : P 99K P . Then how should we compute the inverse Φ−1? The goal of this chapter is to give an answer to this question, provided we have certain data on the presentation of the ideal of the base locus Z. To illustrate the method we propose, we focus on ﬁnding the inverse of the following birational map in characteristic 3 which is the polar map of f = 2 2 2 (x1 + x0x2)x0(x1 + x0x2 + x0).

2 2 Φex : P P 4 3 2 3 3 2 (x0 : x1 : x2) (x1 + x0x2 + x0x1x2 : −x0x1 + x0x1 + x0x1x2 : 4 2 2 3 x0 − x0x1 − x0x2)

n n −1 Recall that if Φ : P 99K P is birational, the inverse Φ is also given by n + 1 0 0 0 homogeneous polynomials φ0, . . . , φn ∈ k[x0, ··· , xn] of the same degree δ such that 0 0 0 0 φ0(φ0, . . . , φn), . . . , φn(φ0, . . . , φn) = (x0P, . . . , xnP ) for a homogeneous polynomial P ∈ k[x0, ··· , xn] and

0 0 φ0(φ0, . . . , φn), . . . , φn(φ0, . . . , φn) = (x0Q, . . . , xnQ) for a homogeneous polynomial Q ∈ k[x0, ··· , xn] what we commonly denote by Φ ◦ Φ−1 = Φ−1 ◦ Φ = id, see Section 1.3. n P We emphasize that if the polynomials φi = aijxj are of degree 1 for all j=0 i ∈ {0, . . . , n}, then Φ is completely deﬁned by the matrix A = (aij)0≤i,j≤n and the problems concerning the birational properties of Φ or concerning the inverse of Φ are equivalent to solve the following linear system   y0 = a00x0 + ... + a0nxn   (E)    yn = an0x0 + ... + annxn

129 130 CHAPTER 7. INVERSE OF A BIRATIONAL MAP that is, Φ is birational if and only if A is invertible and in this case Φ−1 is the map associated to the matrix A−1. So the problem of finding the inverse of birational map can be taken as a generalisation of a classical problem in linear algebra. After presenting in Section 7.1 the currently available techniques to compute Φ−1, we will point out a method based on our analysis of torsion components. The idea is that, while the inverse is essentially known once we have the bigraded generators of the ideal of the Rees algebra (corresponding to the blow-up Γ of X along Z), the computation may fail because these generators are sometimes inac- cessible. On the other hand, the symmetric algebra (corresponding to X = P(IZ )) is much easier to deal with and essentially it is computed from the presentation of IZ . Also, we know that Γ is obtained from X by removing the torsion components, which are supported over the schemes defined by the Fitting ideals of IZ . Then, it suffices to saturate the ideal of X with respect to the Fitting ideals of IZ (pulled back to Pn × Pn) to obtain the desired generators. For the rest of the chapter, the settings are as follows.

n n Notation 7.0.1. Let Φ : P1 99K P2 be a dominant map given by n+1 homogeneous polynomials φ0, . . . , φn ∈ k[x0, ··· , xn] of the same degree δ and without common factor. Let also R1 = k[x0, . . . , xn] (resp. R2 = k[y0, . . . , yn]) be the coordinate n n ring of P1 (resp. P2 ) and denote by IZ the ideal of R1 of the base locus Z of Φ. n n Denote also by S = R1 ⊗R2 the coordinate ring of P1 ×P2 . We identify S with the space of homogeneous polynomials in both variables x0, . . . , xn and y0, . . . , yn and an element P ∈ S of degree d1 in the x variables of degree d2 in the y variables is said of bidegree (d1, d2). −1 n n 0 0 In the case that Φ is birational, we let Φ : P2 99K P1 given by φ0, . . . , φn ∈ k[y0, . . . , yn] be the inverse of Φ with base locus ideal IZ0 .

7.1 State of the art to compute the inverse

Let us adapt ﬁrst the notions and notation of Section 2.1 and Section 2.2 in the context of Notation 7.0.1. Let M be the presentation matrix of IZ and denote by ni n+1 ⊕ R1 (−di) → R1 the graded map deﬁned by M. Our convention is that we do i≥1 0 ni not include pieces of the form R1 in the direct sum ⊕ R1 (−di). i≥1 n n The ideal IX of X = P(IZ ) in P1 × P2 is thus generated by polynomials of bidegree (di, 1) for i such that ni 6= 0. More precisely, there is a surjection

ni ⊕ S (−di, −1) → IX → 0. i≥1

Example 7.1.1. Let φ0, φ1, φ2 be the three polynomials generating the base ideal IZ of Φex. A minimal free resolution of IZ reads

 3 2 2  0 x0 − x0x1 − x0x2 3 M = x0 x1  2 2 x1 −x x2 − x x2 0 1 n+1 0 R1(−1) ⊕ R1(−3) R1 IZ (4) 0 7.1. STATE OF THE ART TO COMPUTE THE INVERSE 131

and IX veriﬁes: S(−1, −1) ⊕ S(−3, −1) → I → 0. P(IZ )

Moreover, as we saw in Section 2.1, the graph ΓΦ of Φ is isomorphic to the n blow-up of P1 along Z such that we have the following commutative diagram

n n P1 × P2

P(IZ ) p1 p2

π1 π2 Γ σ1 σ2

n n P1 P2 Φ where p1 (resp. p2) is the projection over the first (resp. second) factor. Now assume that Φ is birational of inverse Φ−1. Of course Γ = Graph(Φ) = −1 0 Graph(Φ ) but X = P(IZ0 ) might differ from X. This is clear from the study of n n 0 a given set of generator of the ideal I 0 in P1 × P2 . Indeed, as for Φ, let M be the X 0 ni 0 n+1 presentation matrix of IZ0 and denote by ⊕ R2 (−di) → R2 the graded map i≥1 defined by M 0 and there is a surjection

ni 0 ⊕ S (−1, −di) → I (I 0 ) → 0. i≥1 P Z

0 Hence provided that M (or M ) does not have only linear entries, P(IZ ) and P(IZ0 ) n n diﬀer. In this case, they both diﬀer also from Γ since the ideal IΓ of Γ in P1 × P2 contains stricly I and I . P(IZ ) P(IZ0 ) We summarize the situation into the following commutative diagram

n n P1 × P2

0 p1 X X p2

0 π1 Γ π2

σ1 σ2 n n P1 P2 Φ

Φ−1

n where, for the clarity of the diagram, we did not mention the map π2 : X → P2 0 0 n (resp. π1 : X → P1 ) which is the restriction of the second projection p2 to X (resp. 0 of the ﬁrst projection p1 to X ). 132 CHAPTER 7. INVERSE OF A BIRATIONAL MAP

7.1.1 Inverse of the standard Cremona map Recall that the standard Cremona map τ is deﬁned as follows

n n τ : P1 P2 (x0 : ... : xn) (φ0 : ... : φn

where φi = x0 . . . xi−1xi+1 . . . xn for i ∈ {0, . . . , n}. It is a computation to show that the standard Cremona map is an involution i.e. that τ ◦ τ = id. Our goal here is to explain a general method to recover this result. The ideal IZ = (φ0, . . . , φn) of the base locus Z of τ has the following minimal free presentation:

  x0 0 0      −x1 x1     0 −x   2  M =    0 0     x   n−1    0 0 −xn n n+1 0 O n (−1) O n IZ (n) 0 P P

n Hence, denoting y0, . . . , yn the variables of P2 , the ideal sheaf IX of the pro- n n jectivization X of IZ in P1 × P2 is generated by the entries in the row matrix y0 . . . yn M i.e.

IX = (y0x0 − y1x1, . . . , yn−1xn−1 − ynxn).

The remark is that, in this precise case, IX is also generated by the entries of the row matrix:

  y0 0 0    −y1 y1       0 −y2     0 0     yn−1    x0 . . . xn 0 0 −yn

and we can reverse the construction of the projectivization X from the ideal IZ . −1 0 n Indeed, since the inverse τ of τ has also a base locus Z in P2 with ideal IZ0 , 0 n n we can embed the projectivization X of IZ0 into P1 × P2 via the presentation 0 0 0 matrix M of IZ . The ideal IX is generated by the entries in the row matrix 0 x0 . . . xn M . So here a candidate to be the presentation matrix of IZ0 is the matrix: 7.1. STATE OF THE ART TO COMPUTE THE INVERSE 133

  y0 0 0    −y1 y1    0   M = 0 −y2     0 0     yn−1    0 0 −yn and indeed, in the case of the standard map, taking the n × n-minors of M 0, we 0 recover the map τ , up to a change of coordinates i.e. we recover the base ideal IZ0 of τ 0, the inverse of τ. To recover completely τ 0, it remains to identify this last change of coordinates. In order to do it, one solution is to compute n + 1 n × n-minors of M 0 denot- 00 00 00 ed by φ0 , . . . , φn (we denote τ the map associated to these generators of IZ0 ), and to write down explicitly the condition on a change of coordinates associated to a matrix A = (aij)0≤i,j≤n ∈ PGln+1(k). That is equivalent to compute the image y0 = τ(x0), . . . yn+1 = τ(xn+1) of n + 2 general points x0, . . . xn+1 by τ, to compute the image x000 = τ 00(y0), . . . , xn+100 = τ 00(yn+1) and to compute the automorphism of Pn sending the projective basis x000, . . . , xn+100 to the projective basis x0, . . . , xn+1 (this last step, way simpler that what we originally thought of was explained to us by Laurent Manivel).

7.1.2 Jacobian dual Of course the previous example is very special. For instance, it is of linear type and the ideal of the blow-up Γ (or of X) is only generated in bi-degree (1, 1) i.e. is generated by linear forms with respect to the x and y variables. But we give here a sketchy presentation of a result due to F.Russo and A.Simis, extending to more general situations this previous method [RS01]. For the sake of concision and to keep easily the analogy with the standard map, we only present the result in the n n case of a dominant rational map Φ : P 99K P but we emphasize that it extends in greater generality. n n Let Φ : P1 99K P2 be a dominant rational map given by n + 1 homogeneous polynomials φ0, . . . , φn ∈ k[x0, ··· , xn] of degree δ and let

m M n+1 R1 R1 IZ (δ) 0

n be a free presentation of the ideal IZ of the base locus Z of Φ in P . We decompose the presentation matrix M as the concatenation of a matrix M1 whose columns are the columns of M with linear entries and a matrix M2 made by the other n columns of M. We emphasize that, denoting y0, . . . , yn the variables of P2 , the entries of the row matrix y0 . . . yn M1 are the equations (of bi-degree (1, 1)) n n of the projectivization X of IZ in P1 × P2 . Following [RS01], we also denote by Θ the Jacobian matrix of the row matrix n y0 . . . yn M1 with respect to the variables x0, . . . , xn of P1 and q stands for n+1 q the number of columns of M1. Let Z(Θ) be the kernel of Θ : R2 → R2. Given 134 CHAPTER 7. INVERSE OF A BIRATIONAL MAP

a vector g = (g0, . . . , gn), consider the morphism R1 → R2 which sends xi to gi for i ∈ {0, . . . , n − 1} and apply it to the entries of the matrix M1. The result is a matrix with entries in R2 which we will denote by M1(g).

Deﬁnition 7.1.2. The ideal IZ is said to have the strong rank property if the matrix Θ has rank at most n and, for some minimal homogeneous generator g ∈ Z(Θ), the evaluated matrix M1(g) has rank n.

Remark 7.1.3. In Deﬁnition 7.1.2, it is implicit that the strong rank property of an ideal does not depend on the choice of the vector g. The result is even stronger [RS01, Proposition 1.2, Supplement]. It states that any two such vectors g and g0 are proportional.

Looking back to the case of the standard Cremona map, we have that M1 = M, M2 = 0,   y0 0 0    −y1 y1      Θ = 0 −y2     0 0     yn−1    0 0 −yn

and given g = (y1 . . . yn, . . . , y0 . . . yn−1) (which is in Z(Θ)), we have that M1(g) has rank n = rank(M). Hence IZ has the strong rank property and the result is that Φ is birational with an inverse given by the vector g. This is the content of the following proposition which is a combination of [RS01, Theorem 1.4] and [RS01, Proposition 2.1].

n n Proposition 7.1.4. Let Φ: P 99K P be a dominant rational map, M be the presentation matrix of the ideal IZ of the base locus Z of Φ and denote as above by M1 the linear part of M. Then Φ is birational and rank(M) = rank(M1) if and only if IZ has the strong rank property. Moreover, the inverse of Φ is given by any vector g ∈ Z(Θ).

As far as we know, this result is the first to give a computationally very efficient method to construct the inverse of a birational map and this method was indeed the first method implemented in the Macaulay2 package ”Cremona” in order to invert birational maps [Sta17]. However, there exists birational maps Φ whose base locus ideals IZ do not have the strong rank property, for instance Φex. Hence we present now an alternative computational method to invert birational maps.

7.2 Computation of the inverse

We refer to Notation 7.0.1 for all the notation we use and we assume that the map n n Φ: P1 99K P2 is birational. 7.2. COMPUTATION OF THE INVERSE 135

Example 7.2.1. Over a ﬁeld of characteristic diﬀerent from 2, the map Φex is birational but its base locus ideal IZ does not have the strong rank property. Indeed, a presentation of IZ reads

 3 2 2  0 x0 − x0x1 − x0x2 3 x0 x1  2 2 x1 −x0x2 − x1x2 3 0 R1(−1) ⊕ R1(−3) R1 IZ (4) 0. and the matrix M1 of columns with entries of degree 1 of the presentation matrix M of IZ has rank 1 which is smaller than 2. Proposition 7.2.2. The polynomials of bidegree (1, 1) generating I in n × n P(IZ ) P1 P2 are in one-to-one correspondence with the polynomials of bidegree (1, 1) generating I . P(IZ0 )     l0 a00x0 + ... + a0nxn  .   .  Proof. Let  .  =  .  be a linear syzygy of IZ seen as a ln an0x0 + ... + annxn module over R1 generated by (φ0, . . . , φn). Hence, since I ⊂ I , the polynomial y (a x + ... + a x ) + ... + P(IZ ) Graph(Φ) 0 00 0 0n n yn(an0x0 + ... + annxn) is a generator of IGraph(Φ) over S = R1 ⊗ R2. Since

y0(a00x0 + ... + a0nxn) + ... + yn(an0x0 + ... + annxn) =

x0(a00y0 + . . . an0yn) + ... + xn(a0ny0 + ... + annyn)   a00y0 + . . . an0yn −1  .  and that Graph(Φ) = Graph(Φ ), we have that  .  is a (lin- a0ny0 + ... + annyn −1 ear) syzygy of the base locus ideal IZ0 of Φ . Hence, given a minimal presentation of IZ i.e. the presentation matrix M of IZ does not have entry of degree 0, all the columns of degree 1 provide independent 0 columns of degree 1 in the presentation matrix M of IZ0 and there is no other column of degree 1 in M 0 otherwise by the reversibility of the construction, it would provide other columns of degree 1 in M.

If we denote by I the ideal of polynomials of bidegree (1, 1) generating I (1,1) P(IZ ) n n (or I ), the subscheme L = (I ) of × contains both (I ) and (I 0 ) P(IZ0 ) V (1,1) P1 P2 P Z P Z and when IZ has the strong rank property, L and P(IZ ) have the same dimension even if they are not equal. In the case that IZ has the strong rank property, the result is that we can compute the inverse of Φ without computing the equations of Graph(Φ) i.e. without computing the Rees algebra R(IZ ) of IZ . In this perspective, the algorithm we propose for the case that IZ do not have the strong rank property is a method of computing the Rees algebra R(IZ ). We present the algorithm before commenting it.

Algorithm. (1) Compute the presentation matrix M of IZ over R1. 136 CHAPTER 7. INVERSE OF A BIRATIONAL MAP

ai ⊕ R1(−i) M n+1 i≥0 R1 IZ (δ) 0

(2) Over S = R1 ⊗ R2, form the ideal I (I ) generated by the entries in the row P Z matrix y0 . . . yn M.

n−1 ∗ ∞ (3) Compute the saturation ⊕ [I (IZ ) : p1Ii(M) ] of I (IZ ). Since the ideal i=1 P P Ii(M) are the Fitting ideals whose support are the torsion components, the ﬁnal ideal is the ideal IΓ of the blow-up of IZ . Another way to compute IΓ is to compute the primary decomposition of I and to choose the component P(IZ ) of the blow-up Γ. This latter method reveals to be computationally expensive. (4) Form the ideal I generated by the generators of bidegree (1, d0) of I and P(IZ0 ) Γ denote by N the row matrix of those generators. The ideal I is the ideal P(IZ0 ) of P(IZ0 ).

bi N ⊕ S(−1, −i) I i≥0 P(IZ0 ) 0

0 (5) Let M be the jacobian matrix of N with respect to the variables x0, . . . , xn. 0 The matrix M is the presentation matrix IZ0 . (6) Compute the generators of the cokernel. If M 0 is a (n + 1) × n-matrix, this is the same as computing the n × n-minors of M 0. If M 0 is bigger, this is the same as computing the syzygies matrix of the transpose tM 0 of M 0.

0 0 (7) Compute the change of coordinates in order to find generators φ0, . . . , φn of −1 0 0 IZ0 such that Φ = (φ0, . . . , φn) as we explained in the study case of the standard Cremona map. Before commenting the algorithm, we right down its implementation on the computer system Macaulay2 with the map Φex. Example 7.2.3. We give the detailed interaction with Macaulay2. Let us emphasize that we present the computation over a base field of characteristic 3 because it was originally our motivation and the last step of computation of the polynomials 0 0 φ0, . . . , φn is way more easier in this case. (0) We start by the definitions of the field, rings, ideal. Here I stands for the ideal IZ . i1 : k = ZZ/3;

i2 : R1 = k[x_0..x_2];

i3 : R2 = k[y_0..y_2];

i4 : I = ideal (x_1^4+x_0^3*x_2+x_0*x_1^2*x_2,-x_0^3*x_1+ x_0*x_1^3+x_0^2*x_1*x_2, x_0^4-x_0^2*x_1^2-x_0^3*x_2); 7.2. COMPUTATION OF THE INVERSE 137

(1) We compute the presentation matrix M of IZ .

o4 : Ideal of R1

i5 : M = syz gens I

o5 = {4} | 0 x_0^3-x_0x_1^2-x_0^2x_2 | {4} | x_0 x_1^3 | {4} | x_1 -x_0^2x_2-x_1^2x_2 |

3 2 o5 : Matrix R1 <--- R1

(2) We compute now the primary decomposition of I . In the code, we denote P(IZ ) by J the ideal I and by IG the ideal I . P(IZ ) Γ i6 : S = R1**R2; --tensor product of R1 and R2

i7 : J = ideal (matrix{{y_0,y_1,y_2}}*sub(M,S) );

o7 : Ideal of S

i8 : time primaryDecomposition J -- used 0.110233 seconds

2 3 2 o8 = {ideal (x y + x y , x y y + x y + x y y y - x y y 01 12 002 11 1012 212 ------2 3 2 2 2 2 2 -xyy-xy ,xyy -xy -xyy -xxyy -xxy 202 22 002 11 102 0202 022 ------3 2 3 2 2 2 +xxyy,xy -xxy + xy -xxy -xxy -xxy), 1212 00 010 11 020 022 122 ------2 2 ideal (x y + x y , x , x x , x )} 0 1 1 2 1 0 1 0

o8 : List

i9 : IG = (primaryDecomposition J)_0

2 3 2 o9 = ideal (x y + x y , x y y + x y + x y y y - x y y 01 12 002 11 1012 212 138 CHAPTER 7. INVERSE OF A BIRATIONAL MAP

------2 3 2 2 2 2 2 -xyy -xy,xyy -xy -xyy -xxyy -xxy 202 22 002 11 102 0202 022 ------3 2 3 2 2 2 +xxyy,xy -xxy +xy -xxy -xxy -xxy) 1212 00 010 11 020 022 122

o9 : Ideal of S (4) Now, the ideal J0 stands for I . P(IZ0 ) i10 : J’ = ideal ( (gens IG)_(0,0) , (gens IG)_(0,1) );

o10 : Ideal of S (5) We directly right down the matrix M 0.

i11 : M’ = matrix{{y_1,y_0*y_2^2},{y_2,y_1^3+y_0*y_1*y_2}, {0,-y_1^2*y_2-y_0*y_2^2-y_2^3}};

3 2 o11 : Matrix S <--- S

0 and we consider M as a matrix of R2.

0 (6) Even if the ideal IZ0 is the ideal of the 2 × 2-minors of M , we compute the generators of IZ0 with the following more general method: i12 : syz transpose M’

3 1 o12 : Matrix S <--- S

0 where, as for M , we consider the latter matrix with entries in R2. Its entries are the generators of IZ0 .

0 0 (7) For the final step consisting in finding the polynomials φ0, . . . , φn, let us not follow the strategy of finding the change of coordinates by posing the conditions (this strategy amounts to compute the inverse of a 12 × 12 matrix after choosing 4 general points of P2). The fact is that, in characteristic 3, we can find this change of coordinates by testing first some values. And indeed, starting from

00 2 2 3 4 3 2 3 4 2 3 Φ = (−y1y2 − y0y2 − y2 : y1y2 + y0y1y2 + y1y2 : y1 + y0y1y2 − y0y2) 7.2. COMPUTATION OF THE INVERSE 139

we ﬁnd that the inverse of Φ is

−1 2 2 3 4 3 2 3 4 2 3 Φex = (y1y2 + y0y2 + y2 : −y1y2 − y0y1y2 − y1y2 : −y1 − y0y1y2 + y0y2).

0 0 Remark 7.2.4. If the ultimate goal is to ﬁnd explicitly the polynomials φ0 . . . , φn −1 of Φex , we emphasize that the two methods we detailed, namely to pose the conditions over the change of coordinates or to be tricky, require a lot of time. However, as we will see, Item (7) of the algorithm is fastly handled in some Macaulay2 packages (though it is unclear to us what methods are followed by these packages).

Remark 7.2.5. As we explained in Item (3) of the algorithm, computing the primary decomposition of I is equivalent to express the scheme (I ) as the P(IZ ) P Z union of Γ and of other torsion components. But, in our cases of rational maps n n Φ: P 99K P , we know the shape of those torsion components since they are supported over the base scheme Z of Φ. In the case of Φex the base scheme is supported over the point V(x0, x1), hence P(IZ ) is the union of Γ with a torsion 2 part TZ whose reduced structure is isomorphic to P(0:0:1). Hence the ideal of Γ is the saturation of the ideal of P(IZ ) by the ideal ITZ = (x0, x1). The gain is that the computation of the saturation in is less expansive than the computation of the primary decomposition in Item (2). i13 : time saturate(J, ideal(x_0,x_1) ) -- used 0.00715082 seconds

2 3 2 o13 = ideal (x y + x y , x y y + x y + x y y y - x y y 01 12 002 11 1012 212 ------2 3 2 2 2 2 2 -xyy -xy,xyy -xy -xyy -xxyy -xxy 202 22 002 11 102 0202 022 ------3 2 3 2 2 2 +xxyy,xy -xxy +xy -xxy -xxy -xxy) 1212 00 010 11 020 022 122 o13 : Ideal of S

Let us mentions that several Macaulay2 packages, for instance ”Cremona” and ”RationalMaps” packages provide the possibility to compute the inverse of any birational map and not only those with a base ideal having the strong rank property. However, it is not clear from our study of those packages if they use the blow-up algebra and if so, how do they compute it. For us, a problem prolonging the concept of the strong rank property is the following. As we saw in Proposition 7.2.2, the generators of bidegree (1, 1) of the ideal of P(IZ ) are the generators of bidegree (1, 1) of the ideal of P(IZ0 ). Hence, it seems to us that the ideal of P(IZ0 ) should be easier to compute in the case 140 CHAPTER 7. INVERSE OF A BIRATIONAL MAP that I is generated by some polynomials of bidegree (1, 1) or equivalently, if P(IZ ) IZ has some linear syzygies. In this direction, the strong rank property would be the optimal case in which the ideal P(IZ0 ) is the most easily computable. We ﬁnish this chapter with the following remark.

Remark 7.2.6. Recall that, as we explained at the beginning of the chapter, in 2 2 characteristic 3, the map Φex is the polar map of the curve F = V (x1 +x0x2)(x1 + 2 0 x0x2 +x0)x0 . Consider now the dual curve F of F . It is actually the union of the 0 2 0 2 2 dual curve C1 of C1 = V(x1 +x0x2), C2 of V(x1 +x0x2 +x0) and V (y2) the dual of 20 the singular point of F . Here we write y0, y1, y2 for the variables of the dual P . To 0 2 compute C1, we compute the hessian of x1 +x0x2 which is equal in characteristic 3 0 0 1 0 t 2 to H = 0 −1 0. Hence C1 = V y0 y1 y2 H y0 y1 y2 = V(−y1 − 1 0 0 0 2 2 y0y2) and in the same way C2 = V(−y1 − y0y2 + y2). 0 −1 Considering the polar Φex0 of F , we remark that Φex0 = Φex . The fact that the inverse of the polar map is the polar of the dual curve is actually true for all the homaloidal plane curve we know at the moment (i.e. the three homaloidal curves in any characteristic diﬀerent from 2, and the one in characteristic 3). It could be interesting to investigate if this phenomenon is more general. Bibliography

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Dans cette thèse,nous interprétonsgéométriquement la torsion de l’algèbre symétriqued’un faisceau d’idéaux IZ d’un schéma Z définipar n+1 équations dans une variétén-dimensionnelle. Ceci revient àétudierla géométriede la projectivisation de IZ . Les applications de ce point de vue concernent en particulier le domaine des transformations birationnelles de l’espace projectif de dimension 3 au sujet duquel nous construisons des transformations birationnelles explicites qui ont le mêmedegréalgébriqueque leur inverse, le domaine des courbes libres et presque-libres au sujet duquel nous généralisons une caractérisationdes courbes libres en étendant les notions de nombre de Milnor et de nombre de Tjurina. Nous abordons aussi le sujet des hypersurfaces homaloides, notre motivation initiale, au sujet duquel nous exhibons en particulier une courbe homaloide de degré5 en caractéristique3. La dernière application concerne le calcul de l’inverse d’une transformation birationnelle.

Title of the thesis : Geometry of the projectivization of ideals and applications to problems of birationality

Abstract

Keywords: Algebraic Geometry, Commutative algebra, Singularity theory, Bira- tional maps, Homaloidal hypersurfaces, free and nearly free curves, Symmetric and Rees algebra, Syzygies, Resolutions